Chapter 9 Resource Masters - Math Problem Solving · ©Glencoe/McGraw-Hill iv Glencoe Geometry Teacher’s Guide to Using the Chapter 9 Resource Masters The Fast FileChapter Resource

Chapter 9Resource Masters

Geometry

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3

ANSWERS FOR WORKBOOKS The answers for Chapter 9 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-860186-X GeometryChapter 9 Resource Masters

1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03

© Glencoe/McGraw-Hill iii Glencoe Geometry

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Proof Builder . . . . . . . . . . . . . . . . . . . . . . ix

Lesson 9-1Study Guide and Intervention . . . . . . . . 479–480Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 481Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 482Reading to Learn Mathematics . . . . . . . . . . 483Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 484







Chapter 9 AssessmentChapter 9 Test, Form 1 . . . . . . . . . . . . 521–522Chapter 9 Test, Form 2A . . . . . . . . . . . 523–524Chapter 9 Test, Form 2B . . . . . . . . . . . 525–526Chapter 9 Test, Form 2C . . . . . . . . . . . 527–528Chapter 9 Test, Form 2D . . . . . . . . . . . 529–530Chapter 9 Test, Form 3 . . . . . . . . . . . . 531–532Chapter 9 Open-Ended Assessment . . . . . . 533Chapter 9 Vocabulary Test/Review . . . . . . . 534Chapter 9 Quizzes 1 & 2 . . . . . . . . . . . . . . . 535Chapter 9 Quizzes 3 & 4 . . . . . . . . . . . . . . . 536Chapter 9 Mid-Chapter Test . . . . . . . . . . . . 537Chapter 9 Cumulative Review . . . . . . . . . . . 538Chapter 9 Standardized Test Practice . 539–540

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32

© Glencoe/McGraw-Hill iv Glencoe Geometry

Teacher’s Guide to Using theChapter 9 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 9 Resource Masters includes the core materials neededfor Chapter 9. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theGeometry TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 9-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.

Vocabulary Builder Pages ix–xinclude another student study tool thatpresents up to fourteen of the key theoremsand postulates from the chapter. Studentsare to write each theorem or postulate intheir own words, including illustrations ifthey choose to do so. You may suggest thatstudents highlight or star the theorems orpostulates with which they are not familiar.

WHEN TO USE Give these pages tostudents before beginning Lesson 9-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toupdate it as they complete each lesson.

Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

© Glencoe/McGraw-Hill v Glencoe Geometry

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

Assessment OptionsThe assessment masters in the Chapter 9Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 518–519. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

79

© Glencoe/McGraw-Hill vii Glencoe Geometry

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 9.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

component form

dilation

isometry

line of reflection

line of symmetry

point of symmetry

reflection

regular tessellation

resultant

rotation

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

rotational symmetry

scalar

semi-regular tessellation

similarity transformation

standard position

tessellation

transformation

translation

uniform tessellations

vector

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

99

Learning to Read MathematicsProof Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

99

© Glencoe/McGraw-Hill ix Glencoe Geometry

Proo

f Bu

ilderThis is a list of key theorems and postulates you will learn in Chapter 9. As you

study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 9.1

Theorem 9.2

Postulate 9.1

Study Guide and InterventionReflections

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1

© Glencoe/McGraw-Hill 479 Glencoe Geometry

Less

on

9-1

Draw Reflections The transformation called a reflection is a flip of a figure in a point, a line, or a plane. The new figure is the image and the original figure is thepreimage. The preimage and image are congruent, so a reflection is a congruencetransformation or isometry.

Construct the imageof quadrilateral ABCD under areflection in line m .

Draw a perpendicular from each vertexof the quadrilateral to m. Find verticesA�, B�, C�, and D� that are the samedistance from m on the other side of m.The image is A�B�C�D�.

m

C AD

B

A�

B�

C� D�

Quadrilateral DEFG hasvertices D(�2, 3), E(4, 4), F(3, �2), and G(�3, �1). Find the image under reflectionin the x-axis.

To find an image for areflection in the x-axis,use the same x-coordinateand multiply the y-coordinate by �1. Insymbols, (a, b) → (a, �b).The new coordinates areD�(�2, �3), E�(4, �4),F�(3, 2), and G�(�3, 1).The image is D�E�F �G�.

x

y

O

D�E�

F�G�

DE

FG

Example 1Example 1 Example 2Example 2

In Example 2, the notation (a, b) → (a, �b) represents a reflection in the x-axis. Here arethree other common reflections in the coordinate plane.• in the y-axis: (a, b) → (�a, b)• in the line y � x: (a, b) → (b, a)• in the origin: (a, b) → (�a, �b)

Draw the image of each figure under a reflection in line m .

1. 2. 3.

Graph each figure and its image under the given reflection.

4. �DEF with D(�2, �1), E(�1, 3), 5. ABCD with A(1, 4), B(3, 2), C(2, �2),F(3, �1) in the x-axis D(�3, 1) in the y-axis

x

y

OD�

C�

B�

A�

D

C

B

A

x

y

O

F�D�

E�

FD

E

m

Q

R S

TUm

LM

N

OP

m

HJ

K

ExercisesExercises


Lines and Points of Symmetry If a figure has a line of symmetry, then it can befolded along that line so that the two halves match. If a figure has a point of symmetry, itis the midpoint of all segments between the preimage and image points.

Determine how many lines of symmetry a regular hexagon has. Then determine whether a regular hexagon has point symmetry.There are six lines of symmetry, three that are diagonals through opposite vertices and three that are perpendicular bisectors of opposite sides. The hexagon has point symmetrybecause any line through P identifies two points on the hexagon that can be considered images of each other.

Determine how many lines of symmetry each figure has. Then determine whetherthe figure has point symmetry.

1. 2. 3.

4. 5. 6.

7. 8. 9.

A B

C

DE

FP

Study Guide and Intervention (continued)

Reflections

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1

ExampleExample

ExercisesExercises

Skills PracticeReflections

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1


Less

on

9-1

Draw the image of each figure under a reflection in line �.

1. 2.

COORDINATE GEOMETRY Graph each figure and its image under the givenreflection.

3. �ABC with vertices A(�3, 2), B(0, 1), 4. trapezoid DEFG with vertices D(0, �3),and C(�2, �3) in the origin E(1, 3), F(3, 3), and G(4, �3) in the y-axis

5. parallelogram RSTU with vertices 6. square KLMN with vertices K(�1, 0),R(�2, 3), S(2, 4), T(2, �3) and L(�2, 3), M(1, 4), and N(2, 1) in U(�2, �4) in the line y � x the x-axis

Determine how many lines of symmetry each figure has. Then determine whetherthe figure has point symmetry.

7. 8. 9.

M�

K�K

x

y

O

L�

N�

LM

Nx

y

O

R�

S�T�

U�

RS

TU

x

y

O

D

E F

GD�

E�F�

G�

x

y

O

A�B�

C�A

B

C

�

�


Draw the image of each figure under a reflection in line �.

1. 2.

COORDINATE GEOMETRY Graph each figure and its image under the givenreflection.

3. quadrilateral ABCD with vertices 4. �FGH with vertices F(�3, �1), G(0, 4),A(�3, 3), B(1, 4), C(4, 0), and and H(3, �1) in the line y � xD(�3, �3) in the origin

5. rectangle QRST with vertices Q(�3, 2), 6. trapezoid HIJK with vertices H(�2, 5),R(�1, 4), S(2, 1), and T(0, �1) I(2, 5), J(�4, �1), and K(�4, 3) in the x-axis in the y-axis

ROAD SIGNS Determine how many lines of symmetry each sign has. Thendetermine whether the sign has point symmetry.

7. 8. 9.

H I

J

K

H�I�

J�

K�

x

y

OQ�

R�

S�

T�Q

R

S

Tx

y

O

F�

G�

H�

F

G

Hx

y

O

A�B�

C�

D�AB

C

D

x

y

O

��

Practice Reflections

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1

Reading to Learn MathematicsReflections

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1


Less

on

9-1

Pre-Activity Where are reflections found in nature?

Read the introduction to Lesson 9-1 at the top of page 463 in your textbook.

Suppose you draw a line segment connecting a point at the peak of a mountainto its image in the lake. Where will the midpoint of this segment fall?

Reading the Lesson1. Draw the reflected image for each reflection described below.

a. reflection of trapezoid ABCD in the line n b. reflection of �RST in point PLabel the image of ABCD as A�B�C�D�. Label the image of RST as R�S�T�.

c. reflection of pentagon ABCDE in the originLabel the image of ABCDE as A�B�C�D�E�.

2. Determine the image of the given point under the indicated reflection.a. (4, 6); reflection in the y-axisb. (�3, 5); reflection in the x-axisc. (�8, �2); reflection in the line y � xd. (9, �3); reflection in the origin

3. Determine the number of lines of symmetry for each figure described below. Thendetermine whether the figure has point symmetry and indicate this by writing yes or no.a. a square b. an isosceles triangle (not equilateral) c. a regular hexagon d. an isosceles trapezoid e. a rectangle (not a square) f. the letter E

Helping You Remember4. A good way to remember a new geometric term is to relate the word or its parts to

geometric terms you already know. Look up the origins of the two parts of the wordisometry in your dictionary. Explain the meaning of each part and give a term youalready know that shares the origin of that part.

A�B�

C� D�E�

AB

CDE

x

y

O

R�S�

T�

RS P

Tn

A�B�

C�

D�A

BCD


Reflections in the Coordinate Plane

Study the diagram at the right. It shows how the triangle ABC is mapped onto triangle XYZ by thetransformation (x, y) → (�x � 6, y). Notice that �XYZis the reflection image with respect to the vertical line with equation x � 3.

1. Prove that the vertical line with equation x � 3 is theperpendicular bisector of the segment with endpoints (x, y) and (�x � 6, y). (Hint: Use the midpoint formula.)

2. Every transformation of the form (x, y) → (�x � 2h, y) is a reflection with respect to the vertical line with equation x � h. What kind of transformation is (x, y) → ( x, �y � 2 k)?

Draw the transformation image for each figure and the given transformation. Is it a reflection transformation? If so, with respect to what line?

3. (x, y) → (�x � 4, y) 4. (x, y) → (x, �y � 8)

x

y

Ox

y

O

x � 3

A

B

C

X

Y

Z

(x, y) → (�x � 6, y)

x

y

O

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1

Study Guide and InterventionTranslations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2


Less

on

9-2

Translations Using Coordinates A transformation called a translation slides afigure in a given direction. In the coordinate plane, a translation moves every preimagepoint P(x, y) to an image point P(x � a, y � b) for fixed values a and b. In words, atranslation shifts a figure a units horizontally and b units vertically; in symbols,(x, y) → (x � a, y � b).

Rectangle RECT has vertices R(�2, �1),E(�2, 2), C(3, 2), and T(3, �1). Graph RECT and its image for the translation (x, y) → (x � 2, y � 1).The translation moves every point of the preimage right 2 units and down 1 unit.(x, y) → (x � 2, y � 1)R(�2, �1) → R�(�2 � 2, �1 � 1) or R�(0, �2)E(�2, 2) → E�(�2 � 2, 2 � 1) or E�(0, 1)C(3, 2) → C�(3 � 2, 2 � 1) or C�(5, 1)T(3, �1) → T�(3 � 2, �1 � 1) or T�(5, �2)

Graph each figure and its image under the given translation.

1. P�Q� with endpoints P(�1, 3) and Q(2, 2) under the translation left 2 units and up 1 unit

2. �PQR with vertices P(�2, �4), Q(�1, 2), and R(2, 1) under the translation right 2 units and down 2 units

3. square SQUR with vertices S(0, 2), Q(3, 1), U(2, �2), and R(�1, �1) under the translation right 3 units and up 1 unit

x

y

R�

S�Q�

U�R

SQ

U

x

y

P�

Q�

R�

P

QR

x

yP�Q�

PQ

x

y

O

E�

R�

C�

T�

E

R

C

T

ExampleExample

ExercisesExercises


Translations by Repeated Reflections Another way to find the image of atranslation is to reflect the figure twice in parallel lines. This kind of translation is called acomposite of reflections.

In the figure, m || n . Find the translation image of �ABC.�A�B�C� is the image of �ABC reflected in line m .�A�B�C� is the image of �A�B�C� reflected in line n .The final image, �A�B�C�, is a translation of �ABC.

In each figure, m || n . Find the translation image of each figure by reflecting it inline m and then in line n .

1. 2.

3. 4.

5. 6. m

U

A

D

Q Q �

U �

A�

D �

U�

A�

D�

n

m

n

U

T

S

R

U �

T �

S �R �

m

n

D�

Q�

AQ

U

D

A�

Q �

D �

U �T�

m n

P�P P �

E�E E �N N �

T T �

A A�

m n

A�

AB

C

A�

B�

C �

m n

A�

B�

C�

A

B

C

A�

B�

C �

m n

A�

B�

C�

A

B

C

A�

B�

C �


Translations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2

ExercisesExercises

ExampleExample

Skills PracticeTranslations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2


Less

on

9-2

In each figure, a || b. Determine whether figure 3 is a translation image of figure 1.Write yes or no. Explain your answer.

1. 2.

3. 4.

COORDINATE GEOMETRY Graph each figure and its image under the giventranslation.

5. �JKL with vertices J(�4, �4), 6. quadrilateral LMNP with vertices L(4, 2),K(�2, �1), and L(2, �4) under the M(4, �1), N(0, �1), and P(1, 4) under the translation (x, y) → (x � 2, y � 5) translation (x, y) → (x � 4, y � 3)

M�

L

MN

P

L�

N�

P�

x

y

OJ�

K�

L�

J

K

L

x

y

O

a b

12

3

a b1

23

a

b

1

2

3a b

1 2 3


In each figure, c || d . Determine whether figure 3 is a translation image of figure 1.Write yes or no. Explain your answer.

1. 2.

COORDINATE GEOMETRY Graph each figure and its image under the giventranslation.

3. quadrilateral TUWX with vertices 4. pentagon DEFGH with vertices D(�1, �2),T(�1, 1), U(4, 2), W(1, 5), and X(�1, 3) E(2, �1), F(5, �2), G(4, �4), H(1, �4) under the translation under the translation (x, y) → (x � 2, y � 4) (x, y) → (x � 1, y � 5)

ANIMATION Find the translation that moves the figure on the coordinate plane.

5. figure 1 → figure 2



x

y

O

1

3

4

2

D�E�

F�

G�H�

DE

F

GH

x

y

O

W

T�U�

W�

X�

T U

X

x

y

O

c

d

1

2

3

c d

12

3

Practice Translations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2

Reading to Learn MathematicsTranslations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2


Less

on

9-2

Pre-Activity How are translations used in a marching band show?


How do band directors get the marching band to maintain the shape of thefigure they originally formed?

Reading the Lesson1. Underline the correct word or phrase to form a true statement.

a. All reflections and translations are (opposites/isometries/equivalent).b. The preimage and image of a figure under a reflection in a line have

(the same orientation/opposite orientations).c. The preimage and image of a figure under a translation have

(the same orientation/opposite orientations).d. The result of successive reflections over two parallel lines is a

(reflection/rotation/translation).e. Collinearity (is/is not) preserved by translations.f. The translation (x, y) → (x � a, y � b) shifts every point a units

(horizontally/vertically) and y units (horizontally/vertically).

2. Find the image of each preimage under the indicated translation.a. (x, y); 5 units right and 3 units upb. (x, y); 2 units left and 4 units downc. (x, y); 1 unit left and 6 units upd. (x, y); 7 units righte. (4, �3); 3 units upf. (�5, 6); 3 units right and 2 units downg. (�7, 5); 7 units right and 5 units downh. (�9, �2); 12 units right and 6 units down

3. �RST has vertices R(�3, 3), S(0, �2), and T(2, 1). Graph �RSTand its image �R�S�T� under the translation (x, y) → (x � 3, y � 2).List the coordinates of the vertices of the image.

Helping You Remember4. A good way to remember a new mathematical term is to relate it to an everyday meaning

of the same word. How is the meaning of translation in geometry related to the idea oftranslation from one language to another?

R�

S�

T�

R

S

T

x

y

O


Translations in The Coordinate Plane

You can use algebraic descriptions of reflections to show thatthe composite of two reflections with respect to parallel lines isa translation (that is, a slide).

1. Suppose a and b are two different real numbers. Let S and T be thefollowing reflections.

S: (x, y) → (�x � 2 a, y)T: (x, y) → (�x � 2 b, y)

S is reflection with respect to the line with equation x � a, and T isreflection with respect to the line with equation x � b.

a. Find an algebraic description (similar to those above for S andT) to describe the composite transformation “S followed by T.”

b. Find an algebraic description for the composite transformation“T followed by S.”

2. Think about the results you obtained in Exercise 1. What do theytell you about how the distance between two parallel lines isrelated to the distance between a preimage and image point for acomposite of reflections with respect to these lines?

3. Illustrate your answers to Exercises 1 and 2 with sketches. Use aseparate sheet if necessary.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2

Study Guide and InterventionRotations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-39-3


Less

on

9-3

Draw Rotations A transformation called a rotation turns a figure through a specifiedangle about a fixed point called the center of rotation. To find the image of a rotation, oneway is to use a protractor. Another way is to reflect a figure twice, in two intersecting lines.

�ABC has vertices A(2, 1), B(3, 4), and C(5, 1). Draw the image of �ABC under a rotation of 90°counterclockwise about the origin.• First draw �ABC. Then draw a segment from O, the origin,

to point A.• Use a protractor to measure 90° counterclockwise with O�A�

as one side.• Draw OR��.• Use a compass to copy O�A� onto OR��. Name the segment O�A��.• Repeat with segments from the origin to points B and C.

Find the image of �ABC under reflection in lines m and n .First reflect �ABC in line m. Label the image �A�B�C�.Reflect �A�B�C� in line n. Label the image �A�B�C�.�A�B�C� is a rotation of �ABC. The center of rotation is theintersection of lines m and n. The angle of rotation is twice themeasure of the acute angle formed by m and n.

Draw the rotation image of each figure 90° in the given direction about the centerpoint and label the coordinates.

1. P�Q� with endpoints P(�1, �2) 2. �PQR with vertices P(�2, �3), Q(2, �1),and Q(1, 3) counterclockwise and R(3, 2) clockwise about the point T(1, 1)about the origin

Find the rotation image of each figure by reflecting it in line m and then in line n.

3. 4. mn

A B

C

m nQ

P

x

yP�

Q�

R�

P

Q

RT

x

y

P�

Q�

P

Q

m

n

A�

B�C�

A�

B�C�

B

C

A

x

y

O

A�B�

C�

A

B

C

R

ExercisesExercises

Example 1Example 1

Example 2Example 2


Rotational Symmetry When the figure at the right is rotated about point P by 120° or 240°, the image looks like the preimage. The figure has rotational symmetry, which means it can be rotated less than 360° about a point and the preimage and image appear to be the same.

The figure has rotational symmetry of order 3 because there are 3 rotations less than 360° (0°, 120°, 240°) that produce an image that is the same as the original. The magnitude of the rotational symmetry for a figure is 360 degrees divided by the order. For the figure above,the rotational symmetry has magnitude 120 degrees.

Identify the order and magnitude of the rotational symmetry of the design at the right.

The design has rotational symmetry about the center point for rotations of 0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315°.

There are eight rotations less than 360 degrees, so the order of its rotational symmetry is 8. The quotient 360 � 8 is 45, so the magnitude of its rotational symmetry is 45 degrees.

Identify the order and magnitude of the rotational symmetry of each figure.

1. a square 2. a regular 40-gon

3. 4.

5. 6.

P


Rotations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-39-3

ExercisesExercises

ExampleExample

Skills PracticeRotations

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9-39-3


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Rotate each figure about point R under the given angle of rotation and the givendirection. Label the vertices of the rotation image.

1. 90° counterclockwise 2. 90° clockwise

COORDINATE GEOMETRY Draw the rotation image of each figure 90° in the givendirection about the origin and label the coordinates.

3. �STW with vertices S(2, �1), T(5, 1), 4. �DEF with vertices D(�4, 3), E(1, 2),and W(3, 3) counterclockwise and F(�3, �3) clockwise

Use a composition of reflections to find the rotation image with respect to lines kand m. Then find the angle of rotation for each image.

5. 6. k

mL�

M�

N�

O�

LM

NO

k

mA�B�

C�

A

BC

E�

D�F�

F

ED

x

y

OS�

T�

W�

S

T

W

x

y

O

G H

K J R

G�

H�

J�

K�Q

P S

R

Q�

S�

P�


Rotate each figure about point R under the given angle of rotation and the givendirection. Label the vertices of the rotation image.

1. 80° counterclockwise 2. 100° clockwise

COORDINATE GEOMETRY Draw the rotation image of each figure 90° in the givendirection about the center point and label the coordinates.

3. �RST with vertices R(�3, 3), S(2, 4), 4. �HJK with vertices H(3, 1), J(3, �3),and T(1, 2) clockwise about the and K(�3, �4) counterclockwise about point P(1, 0) the point P(�1, �1)

Use a composition of reflections to find the rotation image with respect to lines pand s. Then find the angle of rotation for each image.

5. 6.

7. STEAMBOATS A paddle wheel on a steamboat is driven by a steam engine and movesfrom one paddle to the next to propel the boat through the water. If a paddle wheelconsists of 18 evenly spaced paddles, identify the order and magnitude of its rotationalsymmetry.

p s

P�

R�

S�

T�

P

R

S

T

ps

E�

F�

G�

E

F

G

K�

J�H�

J

H

K

x

y

OP(–1, –1)

S�

R�

T�T

S

R

x

y

O P(1, 0)

QU

T S

P

R

Q�

S�

T� U�

P�

N

M P

R

M�

N�

P�

Practice Rotations

NAME ______________________________________________ DATE ____________ PERIOD _____

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Reading to Learn MathematicsRotations

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Pre-Activity How do some amusement rides illustrate rotations?


What are two ways that each car rotates?

Reading the Lesson

1. List all of the following types of transformations that satisfy each description: reflection,translation, rotation.a. The transformation is an isometry.b. The transformation preserves the orientation of a figure.c. The transformation is the composite of successive reflections over two intersecting

lines.d. The transformation is the composite of successive reflections over two parallel

lines.e. A specific transformation is defined by a fixed point and a specified angle.f. A specific transformation is defined by a fixed point, a fixed line, or a fixed plane.

g. A specific transformation is defined by (x, y) → (x � a, x � b), for fixed values of a and b.

h. The transformation is also called a slide.i. The transformation is also called a flip.j. The transformation is also called a turn.

2. Determine the order and magnitude of the rotational symmetry for each figure.

a. b.

c. d.

Helping You Remember

3. What is an easy way to remember the order and magnitude of the rotational symmetryof a regular polygon?


Finding the Center of RotationSuppose you are told that � X�Y�Z� is the rotation image of � XYZ, but you are not toldwhere the center of rotation is nor the measure of the angle of rotation. Can you find them? Yes,you can. Connect two pairs of correspondingvertices with segments. In the figure, the segments YY� and ZZ� are used. Draw theperpendicular bisectors, � and m, of thesesegments. The point C where � and m intersect is the center of rotation.

1. How can you find the measure of the angle of rotation in the figure above?

Locate the center of rotation for the rotation that maps WXYZ onto W�X�Y�Z�.Then find the measure of the angle of rotation.

2. 3.

Z

Z�

W

W�

X�

X

Y

Y�

center ofrotation

angle ofrotation:about 110

Z�

Z

W

W�X�

X

Y�

Y

center ofrotation

angle ofrotation:about 100

Z�

Z

C

X�X

Y�

Y

m�

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

9-39-3

Study Guide and InterventionTessellations

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9-49-4


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Regular Tessellations A pattern that covers a plane with repeating copies of one ormore figures so that there are no overlapping or empty spaces is a tessellation. A regulartessellation uses only one type of regular polygon. In a tessellation, the sum of the measuresof the angles of the polygons surrounding a vertex is 360. If a regular polygon has an interiorangle that is a factor of 360, then the polygon will tessellate.

regular tessellation tessellation Copies of a regular hexagon Copies of a regular pentagon can form a tessellation. cannot form a tessellation.

Determine whether a regular 16-gon tessellates the plane. Explain.If m�1 is the measure of one interior angle of a regular polygon, then a formula for m�1

is m�1 � �180(n

n� 2)�. Use the formula with n � 16.

m�1 � �180(n

n� 2)�

� �180(1

166

� 2)�

� 157.5

The value 157.5 is not a factor of 360, so the 16-gon will not tessellate.

Determine whether each polygon tessellates the plane. If so, draw a sample figure.

1. scalene right triangle 2. isosceles trapezoid

Determine whether each regular polygon tessellates the plane. Explain.

3. square 4. 20-gon

5. septagon 6. 15-gon

7. octagon 8. pentagon

ExercisesExercises

ExampleExample


Tessellations with Specific Attributes A tessellation pattern can contain any typeof polygon. If the arrangement of shapes and angles at each vertex in the tessellation is thesame, the tessellation is uniform. A semi-regular tessellation is a uniform tessellationthat contains two or more regular polygons.

Determine whether a kite will tessellate the plane. If so, describe the tessellation as uniform, regular, semi-regular, or not uniform.A kite will tessellate the plane. At each vertex the sum of the measures is a � b � b � c, which is 360. The tessellation is uniform.

Determine whether a semi-regular tessellation can be created from each set offigures. If so, sketch the tessellation. Assume that each figure has a side length of1 unit.

1. rhombus, equilateral triangle, 2. square and equilateral triangleand octagon

Determine whether each polygon tessellates the plane. If so, describe thetessellation as uniform, not uniform, regular, or semi-regular.

3. rectangle 4. hexagon and square

c� a�

b�

b�

c�

c�

a�

a�b�

b�

b�

c�a�

b�

b�

c�a�

b�

b�

c�a�

b�

b�

b�

c� a�

b�

b�


Tessellations

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ExercisesExercises

ExampleExample

Skills PracticeTessellations

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1. 15-gon 2. 18-gon

3. square 4. 20-gon

Determine whether a semi-regular tessellation can be created from each set offigures. Assume each figure has a side length of 1 unit.

5. regular pentagons and equilateral triangles

6. regular dodecagons and equilateral triangles

7. regular octagons and equilateral triangles


8. rhombus 9. isosceles trapezoid and square

Determine whether each pattern is a tessellation. If so, describe it as uniform, notuniform, regular, or semi-regular.

10. 11.



1. 22-gon 2. 40-gon

Determine whether a semi-regular tessellation can be created from each set offigures. Assume each figure has a side length of 1 unit.

3. regular pentagons and regular decagons

4. regular dodecagons, regular hexagons, and squares


5. kite 6. octagon and decagon

Determine whether each pattern is a tessellation. If so, describe it as uniform, notuniform, regular, or semi-regular.

7. 8.

FLOOR TILES For Exercises 9 and 10, use the following information.Mr. Martinez chose the pattern of tile shown to retile his kitchen floor.

9. Determine whether the pattern is a tessellation. Explain

10. Is the pattern uniform, regular, or semi-regular?

Practice Tessellations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-49-4

Reading to Learn MathematicsTesselations

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9-4

Pre-Activity How are tessellations used in art?


• In the pattern shown in the picture in your textbook, how many smallequilateral triangles make up one regular hexagon?

• In this pattern, how many fish make up one equilateral triangle?

Reading the Lesson

1. Underline the correct word, phrase, or number to form a true statement.

a. A tessellation is a pattern that covers a plane with the same figure or set of figures sothat there are no (congruent angles/overlapping or empty spaces/right angles).

b. A tessellation that uses only one type of regular polygon is called a(uniform/regular/semi-regular) tessellation.

c. The sum of the measures of the angles at any vertex in any tessellation is(90/180/360).

d. A tessellation that contains the same arrangement of shapes and angles at everyvertex is called a (uniform/regular/semi-regular) tessellation.

e. In a regular tessellation made up of hexagons, there are (3/4/6) hexagons meeting ateach vertex, and the measure of each of the angles at any vertex is (60/90/120).

f. A uniform tessellation formed using two or more regular polygons is called a(rotational/regular/semi-regular) tessellation.

g. In a regular tessellation made up of triangles, there are (3/4/6) triangles meeting ateach vertex, and the measure of each of the angles at any vertex is (30/60/120).

h. If a regular tessellation is made up of quadrilaterals, all of the quadrilaterals must becongruent (rectangles/parallelograms/squares/trapezoids).

2. Write all of the following words that describe each tessellation: uniform, non-uniform,regular, semi-regular.

a. b.

Helping You Remember

3. Often the everyday meanings of a word can help you to remember its mathematicalmeaning. Look up uniform in your dictionary. How can its everyday meanings help youto remember the meaning of a uniform tessellation?


Polygonal NumbersCertain numbers related to regular polygons are called polygonal numbers. The chartshows several triangular, square, and pentagonal numbers. The rank of a polygon numberis the number of dots on each “side” of the outer polygon. For example, the pentagonalnumber 22 has a rank of 4.

Polygonal numbers can be described with formulas. For example, a triangular number T

of rank r can be described by T � �r(r

2� 1)�.

Answer each question.

1. Draw a diagram to find the 2. Draw a diagram to find the triangular number of rank 5. pentagonal number of rank 5.

3. Write a formula for a square number 4. Write a formula for a pentagonalS of rank r. number P of rank r.

5. What is the rank of the pentagonal 6. List the hexagonal numbers for ranksnumber 70? 1 to 5. (Hint: Draw a diagram.)

Rank 1

Triangle

1 3 6 10

1 4 9 16

1 5 12 22

Square

Pentagon

Rank 2 Rank 3 Rank 4

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

9-49-4

Study Guide and InterventionDilations

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9-5

Classify Dilations A dilation is a transformation in which the image may be adifferent size than the preimage. A dilation requires a center point and a scale factor, r.

Let r represent the scale factor of a dilation.

If | r | � 1, then the dilation is an enlargement.

If | r | � 1, then the dilation is a congruence transformation.

If 0 | r | 1, then the dilation is a reduction.

Draw the dilation image of �ABC with center O and r � 2.Draw OA��, OB��, and OC��. Label points A�, B�, and C�so that OA� � 2(OA), OB� � 2(OB), and OC� � 2(OC). �A�B�C� is a dilation of �ABC.

Draw the dilation image of each figure with center C and the given scale factor.Describe each transformation as an enlargement, congruence, or reduction.

1. r � 2 2. r � �12�

3. r � 1 4. r � 3

5. r � �23� 6. r � 1

K J

HF

C

G

M

C

P

R

V

W

M�

P�R�

V�

W�

CS

T S�

T�

C

W

Y

XZ

C

S T

R S� T�

R�C

B

A

A�

B�

O

B

A

CO

B

A

C

A�

B�

C�

ExercisesExercises

ExampleExample


Identify the Scale Factor If you know corresponding measurements for a preimageand its dilation image, you can find the scale factor.

Determine the scale factor for the dilation of X�Y� to A�B�. Determine whether the dilation is an enlargement,reduction, or congruence transformation.

scale factor � �priemimag

aege

lelnegntghth�

� �84

uu

nn

iittss�

� 2The scale factor is greater than 1, so the dilation is an enlargement.

Determine the scale factor for each dilation with center C. Determine whether thedilation is an enlargement, reduction, or congruence transformation.

1. CGHJ is a dilation image of CDEF. 2. �CKL is a dilation image of �CKL.

3. STUVWX is a dilation image 4. �CPQ is a dilation image of �CYZ.of MNOPQR.

5. �EFG is a dilation image of �ABC. 6. �HJK is a dilation image of �HJK.

C

J

K

H

CB

DG

F

EA

C

Z

Y

P

QC

M N

O

PQ

RUX

S T

VW

C

K L

G

C

D

H

F

E

J

A

B

X

Y

C


Dilations

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ExampleExample

ExercisesExercises

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Draw the dilation image of each figure with center C and the given scale factor.

1. r � 2 2. r � �14�

Find the measure of the dilation image M��N�� or of the preimage M�N� using thegiven scale factor.

3. MN � 3, r � 3 4. M�N� � 7, r � 21

COORDINATE GEOMETRY Find the image of each polygon, given the vertices, aftera dilation centered at the origin with a scale factor of 2. Then graph a dilation centered at the origin with a scale factor of �

12�.

5. J(2, 4), K(4, 4), P(3, 2) 6. D(�2, 0), G(0, 2), F(2, �2)

Determine the scale factor for each dilation with center C. Determine whether thedilation is an enlargement, reduction, or congruence transformation. The dashedfigure is the dilation image.

7. 8.

CC

G�

F�

D�

G�

F �

D �x

y

O

J� K�

P�J � K �

P �x

y

O

CC


Draw the dilation image of each figure with center C and the given scale factor.

1. r � �32� 2. r � �

23�

Find the measure of the dilation image A��T�� or of the preimage A�T� using the givenscale factor.

3. AT � 15, r � �35� 4. AT � 30, r � ��

16� 5. A�T� � 12, r � �

43�

COORDINATE GEOMETRY Find the image of each polygon, given the vertices, aftera dilation centered at the origin with a scale factor of 2. Then graph a dilation centered at the origin with a scale factor of �

12�.

6. A(1, 1), C(2, 3), D(4, 2), E(3, 1) 7. Q(�1, �1), R(0, 2), S(2, 1)

Determine the scale factor for each dilation with center C. Determine whether thedilation is an enlargement, reduction, or congruence transformation. The dottedfigure is the dilation image.

8. 9.

10. PHOTOGRAPHY Estebe enlarged a 4-inch by 6-inch photograph by a factor of �52�. What

are the new dimensions of the photograph?

CC

S�

Q�

R�

S �

Q �

R �

x

y

O

C�

D�

E�A�

C �D �

E �A�x

y

O

C

C

Practice Dilations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-59-5

Reading to Learn MathematicsDilations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-59-5


Less

on

9-5

Pre-Activity How do you use dilations when you use a computer?


In addition to the example given in your textbook, give two everydayexamples of scaling an object, one that makes the object larger and anotherthat makes it smaller.

Reading the Lesson1. Each of the values of r given below represents the scale factor for a dilation. In each

case, determine whether the dilation is an enlargement, a reduction, or a congruencetransformation.a. r � 3 b. r � 0.5c. r � �0.75 d. r � �1

e. r � �23� f. r � ��

32�

g. r � �1.01 h. r � 0.999

2. Determine whether each sentence is always, sometimes, or never true. If the sentence isnot always true, explain why.a. A dilation requires a center point and a scale factor.

b. A dilation changes the size of a figure.

c. A dilation changes the shape of a figure.

d. The image of a figure under a dilation lies on the opposite side of the center from thepreimage.

e. A similarity transformation is a congruence transformation.

f. The center of a dilation is its own image.

g. A dilation is an isometry.

h. The scale factor for a dilation is a positive number.

i. Dilations produce similar figures.

Helping You Remember3. A good way to remember something is to explain it to someone else. Suppose that your

classmate Lydia is having trouble understanding the relationship between similaritytransformations and congruence transformations. How can you explain this to her?


Similar CirclesYou may be surprised to learn that two noncongruent circles that lie inthe same plane and have no common interior points can be mappedone onto the other by more than one dilation.

1. Here is diagram that suggests one way to map a smaller circle onto a larger one using a dilation. The circles are given. The linessuggest how to find the center for the dilation. Describe how the center is found.Use segments in the diagram to name the scale factor.

2. Here is another pair of noncongruent circles with no commoninterior point. From Exercise 1, you know you can locate a point off to the left of the smaller circle that is the center for a dilationmapping �C onto �C�. Find another center for another dilationthat maps �C onto �C�. Mark and label segments to name thescale factor.

X

X�

C�CO

A �

M�

M

AO

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

9-59-5

Study Guide and InterventionVectors

NAME ______________________________________________ DATE ____________ PERIOD _____

9-69-6


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on

9-6

Magnitude and Direction A vector is a directed segment representing a quantity that has both magnitude, or length,and direction. For example, the speed and direction of an airplane can be represented by a vector. In symbols, a vector is written as AB�, where A is the initial point and B is the endpoint,or as v�.

A vector in standard position has its initial point at (0, 0) and can be represented by the ordered pair for point B. The vector at the right can be expressed as v� � �5, 3�.

You can use the Distance Formula to find the magnitude | AB� | of a vector. You can describe the direction of a vector bymeasuring the angle that the vector forms with the positivex-axis or with any other horizontal line.

Find the magnitude and direction of AB� for A(5, 2) and B(8, 7).Find the magnitude.

|AB | � �(x2 ��x1)2 �� ( y2 �� y1)2�� (8 � 5�)2 � (�7 � 2�)2�� 34� or about 5.8 units

To find the direction, use the tangent ratio.

tan A � �53� The tangent ratio is opposite over adjacent.

m�A� 59.0 Use a calculator.

The magnitude of the vector is about 5.8 units and its direction is 59°.

Find the magnitude and direction of AB� for the given coordinates. Round to thenearest tenth.

1. A(3, 1), B(�2, 3) 2. A(0, 0), B(�2, 1)

3. A(0, 1), B(3, 5) 4. A(�2, 2), B(3, 1)

5. A(3, 4), B(0, 0) 6. A(4, 2), B(0, 3)

x

y

O

B(8,7)

A(5, 2)

x

y

O

B(5, 3)

A(0, 0)

V�

ExercisesExercises

ExampleExample


Translations with Vectors Recall that the transformation (a, b) → (a � 2, b � 3)represents a translation right 2 units and down 3 units. The vector �2, �3� is another way todescribe that translation. Also, two vectors can be added: �a, b� � �c, d� � �a � c, b � d�. Thesum of two vectors is called the resultant.

Graph the image of parallelogram RSTUunder the translation by the vectors m� � �3, �1� and n� � ��2, �4�.Find the sum of the vectors.m�� n� � �3, �1� � ��2, �4�

� �3 � 2, �1 � 4�� 1, �5�

Translate each vertex of parallelogram RSTU right 1 unit and down 5 units.

Graph the image of each figure under a translation by the given vector(s).

1. �ABC with vertices A(�1, 2), B(0, 0), 2. ABCD with vertices A(�4, 1), B(�2, 3),and C(2, 3); m�� 2, �3� C(1, 1), and D(�1, �1); n� � �3 �3�

3. ABCD with vertices A(�3, 3), B(1, 3), C(1, 1), and D(�3, 1); the sum of p� � ��2, 1� and q� � �5, �4�

Given m� � �1, �2� and n� � ��3, �4�, represent each of the following as a singlevector.

4. m�� n� 5. n� � m�

x

y

D�

A�B�

C�

D

A B

C

x

y

D�

A�

B�

C�D

A

B

C

x

y

A�

B�

C�

A

B

C

x

y

R�S�

T�

U�

R

S

TU


Vectors

NAME ______________________________________________ DATE ____________ PERIOD _____

9-69-6

ExampleExample

ExercisesExercises

Skills PracticeVectors

NAME ______________________________________________ DATE ____________ PERIOD _____

9-69-6


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Write the component form of each vector.

1. 2.

Find the magnitude and direction of RS� for the given coordinates. Round to thenearest tenth.

3. R(2, �3), S(4, 9) 4. R(0, 2), S(3, 12)

5. R(5, 4), S(�3, 1) 6. R(1, 5), S(�4, �6)


7. �ABC with vertices A(�4, 3), B(�1, 4), 8. trapezoid with vertices T(�4, �2),C(�1, 1); t� � �4, �3� R(�1, �2), S(�2, �3), Y(�3, �3); a� � �3, 1�

and b� � �2, 4�

Find the magnitude and direction of each resultant for the given vectors.

9. y� � �7, 0�, z� � �0, 6� 10. b� � �3, 2�, c� � ��2, 3�

S�Y�

T� R�

SY

T Rx

y

O

C�

B�A�C

B

A

x

y

O

x

y

O

B(–2, 2)

D(3, –3)

x

y

OC(1, –1)

E(4, 3)


Write the component form of each vector.

1. 2.

Find the magnitude and direction of FG� for the given coordinates. Round to thenearest tenth.

3. F(�8, �5), G(�2, 7) 4. F(�4, 1), G(5, �6)


5. �QRT with vertices Q(�1, 1), R(1, 4), 6. trapezoid with vertices J(�4, �1),T(5, 1); s� � ��2, �5� K(0, �1), L(�1, �3), M(�2, �3); c� � �5, 4�

and d� � ��2, 1�

Find the magnitude and direction of each resultant for the given vectors.

7. a� � ��6, 4�, b� � �4, 6� 8. e� � ��4, �5�, f� � ��1, 3�

AVIATION For Exercises 9 and 10, use the following information.A jet begins a flight along a path due north at 300 miles per hour. A wind is blowing due westat 30 miles per hour.

9. Find the resultant velocity of the plane.

10. Find the resultant direction of the plane.

L�M�

J� K�

LM

JK

x

y

O

T�

R�

Q�

T

R

Qx

y

O

x

y

O

K(–2, 4)

L(3, –4)

x

y

O

A(–2, –2)

B(4, 2)

Practice Vectors

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Pre-Activity How do vectors help a pilot plan a flight?


Why do pilots often head their planes in a slightly different direction fromtheir destination?

Reading the Lesson1. Supply the missing words or phrases to complete the following sentences.

a. A is a directed segment representing a quantity that has bothmagnitude and direction.

b. The length of a vector is called its .

c. Two vectors are parallel if and only if they have the same or direction.

d. A vector is in if it is drawn with initial point at the origin.

e. Two vectors are equal if and only if they have the same and the

same .

f. The sum of two vectors is called the .

g. A vector is written in if it is expressed as an ordered pair.

h. The process of multiplying a vector by a constant is called .

2. Write each vector described below in component form.

a. a vector in standard position with endpoint (a, b)

b. a vector with initial point (a, b) and endpoint (c, d)

c. a vector in standard position with endpoint (�3, 5)

d. a vector with initial point (2, �3) and endpoint (6, �8)

e. a� � b� if a� � ��3, 5� and b� � �6, �4�

f. 5u� if u� � �8, �6�

g. ��13�v� if v� � ��15, 24�

h. 0.5u� � 1.5v� if u� � �10, �10� and v� � ��8, 6�

Helping You Remember3. A good way to remember a new mathematical term is to relate it to a term you already

know. You learned about scale factors when you studied similarity and dilations. How is theidea of a scalar related to scale factors?


Reading MathematicsMany quantities in nature can be thought of as vectors. The science ofphysics involves many vector quantities. In reading about applicationsof mathematics, ask yourself whether the quantities involve onlymagnitude or both magnitude and direction. The first kind of quantityis called scalar. The second kind is a vector.

Classify each of the following. Write scalar or vector.

1. the mass of a book

2. a car traveling north at 55 mph

3. a balloon rising 24 feet per minute

4. the size of a shoe

5. a room temperature of 22 degrees Celsius

6. a west wind of 15 mph

7. the batting average of a baseball player

8. a car traveling 60 mph

9. a rock falling at 10 mph

10. your age

11. the force of Earth’s gravity acting on a moving satellite

12. the area of a record rotating on a turntable

13. the length of a vector in the coordinate plane

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Translations and Dilations A vector can be represented by the ordered pair �x, y� or

by the column matrix . When the ordered pairs for all the vertices of a polygon are

placed together, the resulting matrix is called the vertex matrix for the polygon.

For �ABC with A(�2, 2), B(2, 1), and C(�1, �1), the vertex

matrix for the triangle is .

For �ABC above, use a matrix to find the coordinates of thevertices of the image of �ABC under the translation (x, y) → (x � 3, y � 1).To translate the figure 3 units to the right, add 3 to each x-coordinate. To translate thefigure 1 unit down, add �1 to each y-coordinate.

Vertex Matrix Translation Vertex Matrixof �ABC Matrix of �A�B�C�

� �

The coordinates are A�(1, 1), B�(5, 0), and C�(2, �2).

For �ABC above, use a matrix to find the coordinates of the verticesof the image of �ABC for a dilation centered at the origin with scale factor 3.

Scale Vertex Matrix Vertex MatrixFactor of �ABC of �A�B�C�

3 �

The coordinates are A�(�6, 6), B�(6, 3), and C�(�3, �3).

Use a matrix to find the coordinates of the vertices of the image of each figureunder the given translations or dilations.

1. �ABC with A(3, 1), B(�2, 4), C(�2, �1); (x, y) → (x � 1, y � 2)

2. parallelogram RSTU with R(�4, �2), S(�3, 1), T(3, 4), U(2, 1); (x, y) → (x � 4, y � 3)

3. rectangle PQRS with P(4, 0), Q(3, �3), R(�3, �1), S(�2, 2); (x, y) → (x � 2, y � 1)

4. �ABC with A(�2, �1), B(�2, �3), C(2, �1); dilation centered at the origin with scalefactor 2

5. parallelogram RSTU with R(4, �2), S(�4, �1), T(�2, 3), U(6, 2); dilation centered at theorigin with scale factor 1.5

�6 6 �3 6 3 �3

�2 2 �1 2 1 �1

1 5 21 0 �2

3 3 3�1 �1 �1

�2 2 �1 2 1 �1

�2 2 �1 2 1 �1

x

y

B(2, 1)

C(–1, –1)

A(–2, 2)

x y

ExercisesExercises

Example 1Example 1

Example 2Example 2


Reflections and Rotations When you reflect an image, one way to find thecoordinates of the reflected vertices is to multiply the vertex matrix of the object by areflection matrix. To perform more than one reflection, multiply by one reflection matrixto find the first image. Then multiply by the second matrix to find the final image. Thematrices for reflections in the axes, the origin, and the line y � x are shown below.

For a reflection in the: x-axis y-axis origin line y � x

Multiply the vertex matrix by:

�ABC has vertices A(�2, 3), B(1, 4), and C(3, 0). Use a matrix to find the coordinates of the vertices of the image of �ABC after a reflection in the x-axis.To reflect in the x-axis, multiply the vertex matrix of �ABC by thereflection matrix for the x-axis.

Reflection Matrix Vertex Matrix Vertex Matrix for x-axis of �ABC of �A’B’C’

�

Use a matrix to find the coordinates of the vertices of the image of each figureunder the given reflection.

1. �ABC with A(�3, 2), B(�1, 3), C(1, 0); reflection in the x-axis

2. �XYZ with X(2, �1), Y(4, �3), Z(�2, 1); reflection in the y-axis

3. �ABC with A(3, 4), B(�1, 0), C(�2, 4); reflection in the origin

4. parallelogram RSTU with R(�3, 2), S(3, 2), T(5, �1), U(�1, �1); reflection in the line y � x

5. �ABC with A(2, 3), B(�1, 2), C(1, �1); reflection in the origin, then reflection in the liney � x

6. parallelogram RSTU with R(0, 2), S(4, 2), T(3, �2), U(�1, �2); reflection in the x-axis,then reflection in the y-axis

�2 1 3�3 �4 0

�2 1 3 3 4 0

1 00 �1

x

y

A�B�

AB

C

0 11 0

�1 0 0 �1

�1 0 0 1

1 00 �1


Transformations with Matrices

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ExercisesExercises

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Use a matrix to find the coordinates of the vertices of the image of each figureunder the given translations.

1. �STU with S(6, 4), T(9, 7), and U(14, 2); (x, y) → (x � 4, y � 3)

2. �GHI with G(�5, 0), H(�3, 6), and I(�2, 1); (x, y) → (x � 2, y � 6)

Use scalar multiplication to find the coordinates of the vertices of each figure fora dilation centered at the origin with the given scale factor.

3. �DEF with D(2, 1), E(5, 4), and F(7, 2); r � 4

4. quadrilateral WXYZ with W(�9, 6), X(�6, 3), Y(3, 12), and Z(�6, 15); r � �13�


5. �MNO with M(�5, 1), N(�2, 3), and O(2, 0); y-axis

6. quadrilateral ABCD with A(3, 1), B(6, �2), C(5, �5), and D(1, �6); x-axis

Use a matrix to find the coordinates of the vertices of the image of each figureunder the given rotation.

7. �RST with R(�2, �2), S(�3, 3), and T(2, 2); 90°counterclockwise

8. �LMNP with L(3, 4), M(7, 4), N(9, �3), and P(5, �3); 180° counterclockwise


Use a matrix to find the coordinates of the vertices of the image of each figureunder the given translations.

1. �KLM with K(�7, �3), L(4, 9), and M(9, �6); (x, y) → (x � 7, y � 2)

2. �ABCD with A(�4, 3), B(�2, 8), C(3, 10), and D(1, 5); (x, y) → (x � 3, y � 9)

Use scalar multiplication to find the coordinates of the vertices of each figure fora dilation centered at the origin with the given scale factor.

3. quadrilateral HIJK with H(�2, 3), I(2, 6), J(8, 3), and K(3, �4); r � ��13�

4. pentagon DEFGH with D(�8, �4), E(�8, 2), F(2, 6), G(8, 0), and H(4, �6); r � �54�


5. �QRS with Q(�5, �4), R(�1, �1), and S(2, �6); x-axis

6. quadrilateral VXYZ with V(�4, �2), X(�3, 4), Y(2, 1), and Z(4, �3); y � x

Use a matrix to find the coordinates of the vertices of the image of each figureunder the given rotation.

7. �EFGH with E(�5 �4), F(�3, �1), G(5, �1), and H(3, �4); 90° counterclockwise

8. quadrilateral PSTU with P(�3, 5), S(2, 6), T(8, 1), and U(�6, �4); 270° counterclockwise

9. FORESTRY A research botanist mapped a section of forested land on a coordinate gridto keep track of endangered plants in the region. The vertices of the map are A(�2, 6),B(9, 8), C(14, 4), and D(1, �1). After a month, the botanist has decided to decrease the

research area to �34� of its original size. If the center for the reduction is O(0, 0),what are

the coordinates of the new research area?

Practice Transformations with Matrices

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Reading to Learn MathematicsTransformations with Matrices

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Less

on

9-7

Pre-Activity How can matrices be used to make movies?


• What kind of transformation should be used to move a polygon?

• What kind of transformation should be used to resize a polygon?

Reading the Lesson1. Write a vertex matrix for each figure.

a. �ABC b. parallelogram ABCD

2. Match each transformation from the first column with the corresponding matrix fromthe second or third column. In each case, the vertex matrix for the preimage of a figureis multiplied on the left by one of the matrices below to obtain the image of the figure.All rotations listed are counterclockwise through the origin. (Some matrices may be usedmore than once or not at all.)

a. reflection over the y-axis i. v.b. 90° rotationc. reflection over the line y � x ii. vi.d. 270° rotatione. reflection over the origin iii. vii.f. 180° rotationg. reflection over the x-axis iv. viii.h. 360° rotation

Helping You Remember3. How can you remember or quickly figure out the matrices for the transformations in

Exercise 2?

0 �11 0

0 �1�1 0

1 00 �1

0 11 0

�1 0 0 1

�1 0 0 �1

0 1�1 0

1 00 1

B C

A D x

y

O

B

C

A

x

y

O


Vector AdditionVectors are physical quantities with magnitude and direction. Force and velocity are twoexamples. We will investigate adding vector quantities. The sum of two vectors is called aresultant vector or just the resultant.

Two separate forces, one measuring 20 units and the other measuring 40 units, act on an object. If the angle between the forces is 50°, find themagnitude and direction of the resultant force.

First, the vectors must be rearranged by placing the tail of the 20-unit vector at the head of the 40-unit vector. Since these vectors are not perpendicular, the horizontal and verticalcomponents of one of the vectors must be found. Using trigonometry, the horizontal component must be (20 cos 50°) units and the vertical component must be (20 sin 50°) units.Replacing the 20-unit vector with these components, we can now form two vectors perpendicular and use the Pythagorean Theorem to find the resultant.

r2 � (40 � 20 cos 50°)2 � (20 sin 50°)2

r2 � (52.9)2 � (15.3)2

r2 � 3032.5r2 � 55.1

tan O � �402�0

2s0in

c5os0°

50°�

� 0.2898m�O � 16

Therefore, the resultant force is 55.1 units directed 16° from the 40-unit force.

Solve. Round all angle measures to the nearest degree. Round all other measuresto the nearest tenth.

1. A plane flies due west at 250 kilometers per hour while the wind blows south at 70kilometers per hour. Find the plane’s resultant velocity.

2. A plane flies east for 200 km, then 60° south of east for 80 km. Find the plane’s distanceand direction from its starting point.

3. One force of 100 units acts on an object. Another force of 80 units acts on the object at a40° angle from the first force. Find the resultant force on the object.

O

20

40 � 20 cos 50�

20 sin 50�r

20

20 cos 50�

20 sin 50�

50�

O

20 units

40 units50�

O

20 units

40 units50�

Enrichment

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ExampleExample

Chapter 9 Test, Form 199


Ass

essm

ents

Write the letter for the correct answer in the blank at the right of eachquestion.

1. Given A(3, �7), under which reflection is A�(3, 7)?A. reflection in the x-axis B. reflection in the y-axisC. reflection in the origin D. reflection in y � x

2. Name the image of B�C� under reflection in line m.A. B�C� B. B�A�C. A�C� D. line �

3. How many lines of symmetry does a square have?A. 0 B. 2 C. 4 D. 8

4. Which of the following will result in a translation?A. reflecting in two parallel linesB. reflecting in two intersecting linesC. reflecting in two perpendicular linesD. turning the figure upside down

5. Which transformation moves all points the same distance in the samedirection?A. rotation B. translationC. reflection D. dilation

6. What is the image of X(3, 5) under the translation (x, y) → (x � 4, y � 6)?A. X�(7, �1) B. X�(�1, �1)C. X�(7, 11) D. X�(�1, 11)

7. Find the measure of the angle between two intersecting lines if successivereflections in these lines creates a rotation of 80°.A. 180 B. 160 C. 80 D. 40

8. Find the angle of rotation if the preimage is reflected in perpendicular lines.A. 45° B. 90° C. 180° D. 360°

9. Which figure could tessellate the plane?A. regular pentagon B. regular hexagonC. regular octagon D. regular heptagon

10. Which term describes this tessellation?A. regular B. semi-regularC. not uniform D. uniform

11. What type of dilation occurs with a scale factor of �32�?

A. enlargement B. reductionC. congruence transformation D. inverse transformation

B

CA�

m

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 9 Test, Form 1 (continued)99

12.

13.

14.

15.

16.

17.

18.

19.

20.

12. If �A�B�C� is the image of �ABC under a dilation with center at (0, 0). Find the scale factor.

A. 3 B. �23�

C. �13� D. ��

13�

13. Joe’s old graphing calculator had 96 pixels across the screen. His newcalculator has 144 pixels. Find the scale factor by which he increased hisscreen size.

A. �12� B. �

23� C. �

32� D. 48

14. Find the component form of AB� with A(2, 3) and B(�4, 6).A. ��2, 9� B. �2, �9� C. ��6, 3� D. �6, �3�

15. Find the magnitude of AB� with A(3, 4) and B(�1, 7).A. �4, �3� B. 5 C. �13� D. 25

16. Find the direction of AB� with A(3, 4) and B(�1, 7) to the nearest tenth.A. 36.9° B. 53.1° C. 126.9° D. 143.1°

17. Find the image of A(3, 7) under a translation by a� � ��4, 2�.A. A�(�7, �5) B. A�(�1, 9) C. A�(7, 5) D. A�(1, �9)

18. Which translation matrix could be used to translate �ABC 5 units to theright and 7 units up?

A. B. C. D.

19. Find the coordinates of X� with X(6, 5) for a dilation centered at the originwith a scale factor of �2.A. X�(�10, �12) B. X�(10, 12) C. X�(12, 10) D. X�(�12, �10)

20. Which reflection matrix could you use to reflect a figure in the x-axis?

A. B. C. D.

Bonus Describe a rotation that movestriangle 1 to triangle 2.

1 2

0 11 0

�1 0 0 �1

�1 0 0 1

1 00 �1

�5 �5 �5�7 �7 �7

5 5 5�7 �7 �7

�5 �5 �5 7 7 7

5 5 57 7 7

x

y

O

A�

B� C�

A

B C

B:

NAME DATE PERIOD

Chapter 9 Test, Form 2A99


Ass

essm

ents


1. Given B(�4, �6), under which reflection is B�(4, 6)?A. reflected in the x-axis B. reflected in the y-axisC. reflected in the origin D. reflected in y � x

2. Name the image of E�F� under reflection in line �.A. F�G� B. H�G�C. E�H� D. F�E�

3. How many lines of symmetry does a regular decagon have?A. 0 B. 2 C. 5 D. 10

4. Which property is changed by a translation?A. collinearity B. angle measureC. distance measure D. position

5. What is the image of Y(�4, 7) under the translation (x, y) → (x � 3, y � 5)?A. Y�(�1, 2) B. Y�(�1, 12) C. Y�(�7, 2) D. Y�(�7, 12)

6. What is a transformation called that turns every point of the preimagethrough a specified angle and direction about a fixed point?A. reflection B. rotation C. translation D. dilation

7. Find the angle of rotation for a figure reflected in two lines that intersect toform a 72° angle.A. 36° B. 72° C. 144° D. 288°

8. Find the sum of the measures of the angles of the polygons in a tessellationat a vertex.A. 90 B. 180 C. 360 D. 720

9. Describe this tessellation.A. uniform and regularB. uniform and semi-regularC. not uniform but regularD. not uniform and semi-regular

10. What type of dilation occurs with a scale factor of �14�?

A. enlargement B. reductionC. congruence transformation D. inverse transformation

E

G

F

H

�

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 9 Test, Form 2A (continued)99

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

11. Find the scale factor if �D�E�F� is the image of �DEF under a dilation with center C.A. 2 B. 1C. �1 D. �2

12. Sue scans a 4-inch picture into her computer. She stretches the picture’slength to 10 inches. Find the scale factor she used.

A. 6 B. �52� C. 2 D. �

25�

13. Find the component form of CD� with C(5, �7) and D(�3, 9).A. ��2, 2� B. �2, 2� C. �8, �16� D. ��8, 16�

14. Find the magnitude of CD� with C(5, 2) and D(�1, 6).A. ��6, 4� B. 2�13� C. 4�5� D. 10

15. Find the direction of CD� with C(5, 2) and D(�1, 6) to the nearest tenth.A. 33.7° B. 56.3° C. 123.7° D. 146.3°

16. Find the image of P(�2, 4) under a translation by the vector b� � �6, 5�.A. P�(4, 9) B. P�(�4, �9) C. P�(�8, �1) D. P�(8, 1)

17. HIJK is a trapezoid with H(5, 4), I(10, �2), J(�8, �2), and K(�3, 4). Find thecoordinates of the image of H under (x, y) → (x � 10, y � 11).A. (20, �13) B. (15, �7) C. (�5, 15) D. (7, �7)

18. The vertex matrix of a figure is multiplied on the left by . Find theangle of rotation about the origin.A. 90° counterclockwise B. 180° counterclockwiseC. 270° counterclockwise D. 360° counterclockwise

19. Find the matrix that is used to rotate a figure 270° counterclockwise aboutthe origin.

A. B. C. D.

20. Find the matrix you could use to reflect a figure in the origin.

A. B. C. D.

Bonus Find a vector in component form with magnitude 1 in a direction opposite to a� � ��3, �4�.

0 11 0

�1 0 0 �1

1 00 �1

�1 0 0 1

0 �1�1 0

0 1�1 0

�1 0 0 �1

0 �11 0

0 �11 0

x

y

O

E�

D�

F�D

F E

C

B:

NAME DATE PERIOD

Chapter 9 Test, Form 2B99


Ass

essm

ents


1. Given C(2, 9), under which reflection is C�(9, 2)?A. reflected in the x-axis B. reflected in the y-axisC. reflected in the origin D. reflected in y � x

2. Name the image of K�L� under reflection in point N.A. L�K� B. M�L�C. M�J� D. J�K�

3. How many lines of symmetry does a regular 12-gon have?A. 4 B. 6 C. 12 D. 24

4. Which of the following figures shows a translation?A. B.

C. D.

5. Name the image of C(6, �4) under rotation 90° counterclockwise about theorigin.A. C�(4, 6) B. C�(�4, �6) C. C�(6, 4) D. C�(�6, �4)

6. Name the image of Z(�11, �6) under the translation (x, y) → (x � 1, y � 7).A. Z�(�12, 1) B. Z�(�12, �13) C. Z�(�10, 1) D. Z�(�10, �13)

7. Find the angle of rotation of a figure that is reflected in two lines thatintersect to form a 66° angle.A. 264° B. 132° C. 66° D. 33°

8. Which could be used to create a regular tessellation?A. square B. rectangle C. trapezoid D. all of these

9. Describe this tessellation.A. uniform and regularB. uniform and semi-regularC. not uniform but regularD. not uniform and semi-regular

10. What type of dilation occurs with a scale factor of 1?A. enlargement B. reductionC. congruence transformation D. inverse transformation

J

N

K

M L

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 9 Test, Form 2B (continued)99

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

11. Find the scale factor if �X�Y�Z� is the image of �XYZ under a dilation with center C.A. 6 B. 3

C. 2 D. �13�

12. Stella carved an equilateral triangle that is 3 centimeters on each side of apumpkin. Then she carved another equilateral triangle that is 5 centimeters oneach side. What scale factor did Stella use to increase the size of the triangle?

A. 2 B. �53� C. �

23� D. �

35�

13. Find the component form of EF� with E(�11, �3) and F(7, �4).A. ��18, 1� B. �18, �1� C. ��4, �7� D. �4, 7�

14. Find the magnitude of GH� with G(�3, 0) and H(4, 5).A. ��7, �5� B. �7, 5� C. �26� D. �74�

15. Find the direction of GH� with G(�3, 0) and H(4, 5) to the nearest tenth.A. 35.5° B. 54.5° C. 125.5° D. 144.5°

16. Find the image of D(3, 4) under a translation by the vector c� � ��7, �2�.A. D�(10, 6) B. D�(�10, �6) C. D�(4, �2) D. D�(�4, 2)

17. Find the image Y(�7, 4) for a dilation centered at the origin with a scalefactor of �3.A. Y�(�21, 12) B. Y�(21, �12) C. Y�(12, �21) D. Y�(�12, 21)

18. LMNO is a trapezoid with L(�1, 4), M(5, 12), N(5, �3), and O(�1, �5). Findthe coordinates for the image of N under the translation (x, y) → (x � 7, y � 9).A. (�2, 6) B. (12, �12) C. (�8, 13) D. (�2, 21)

19. If you multiply a vertex matrix of a figure on the left by , under a

rotation counterclockwise about the origin, of what magnitude will you findthe vertices of the image?A. 90° B. 180° C. 270° D. 360°

20. To reflect a figure in the line y � x you can multiply a vertex matrix of afigure on the left by which of the following matrices?

A. B. C. D.

Bonus Find a vector in component form in the same direction as ��3, �4� with magnitude 15.

0 11 0

�1 0 0 �1

1 00 �1

�1 0 0 1

�1 0 0 �1

x

y

OX�Z�

Y�

Z

C

XY

B:

NAME DATE PERIOD

Chapter 9 Test, Form 2C99


Ass

essm

ents

1. Write the coordinates of the image of P(�2, 5) reflected in theline y � x.

2. Graph �ABC with vertices A(4, 4), B(3, �2), and C(�1, �1).Then graph the image of �ABC reflected in the y-axis.

3. How many lines of symmetry does this figure have?

4. Determine whether �A�B�C� is a translation image of �ABC. Explainyour answer.

5. Find the image of W�X� with W(7, 1) and X(�4, 5) under thetranslation (x, y) → (x � 4, y � 3).

6. Find the image of A�B� with A(�3, 1) and B(�1, 5) under arotation of 90° clockwise about the origin.

7. Find the coordinates of L� if �LMN with L(3, 1), M(�1, 6), andN(�3, 2) is reflected in the line y � x and then in the x-axis.

8. Determine whether a regular 12-gon tessellates the plane.Explain.

9. Describe this tessellation as uniform,not uniform, regular, or semi-regular.

10. If AB � 10 and A�B� � 5, is the dilation an enlargement,reduction, or congruence transformation?

aA�

B�C�

A

BC

A�

B �C �

b

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

A�

B�

C�

A

B

C

NAME DATE PERIOD

SCORE


Chapter 9 Test, Form 2C (continued)99

11. Find the measure of the image of S�T� if ST � 4 under a

dilation with a scale factor of �34�.

12. Draw the image of �MNO under a dilation with center C anda scale factor of 2.

13. If ST � 12 and S�T� � 9, find the scale factor of the dilation.

14. Find the direction of n� � ��7, �3� to the nearest tenth.

15. Find the magnitude of k� � �6, 8�.

16. Write the component form of AB�.

17. If a� and m� have opposite directions, are they parallel?

18. Find the image of the point at (5, 1) under the translation bym�� 9, 6�.

19. Use a matrix to find the coordinates of the vertices of theimage of �EFG with E(6, 1), F(�1, �3), and G(2, 4), under thetranslation (x, y) → (x � 3, y � 2).

20. Use a matrix to find the coordinates of the vertices of theimage of �ABC with A(0, 2), B(3, 6), and C(5, 0), after areflection in the y-axis.

Bonus An airplane is flying at 400 miles per hour due west. Thewind is blowing from due north at 30 miles per hour. Findthe resultant speed and direction of the plane.

x

y

O

A

B

NAME DATE PERIOD

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12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

M�

N�

O�M

N

OC

Chapter 9 Test, Form 2D99


Ass

essm

ents

1. Write the coordinates of the image of Q(�3, �6) reflected inthe origin.

2. Graph �PQR with vertices P(3, 4), Q(5, �1), and R(�3, 0).Then graph the image of �PQR reflected in the x-axis.

3. How many lines of symmetry does this figure have?

4. Determine whether W�X�Y�Z� is a translation image of WXYZ. Explain.

5. Find the image of U�V� with U(�3, 5) and V(0, 8) under thetranslation (x, y) → (x � 2, y � 5).

6. Find the image of C�D� with C(0, 4) and D(3, 4) under a rotationof 90° counterclockwise about the origin.

7. Find the coordinates of Q� if �OPQ with O(4, 2), P(5, 0), andQ(1, �2) is reflected in the x-axis and then in the y-axis.

8. Determine whether a regular 15-gon tessellates the plane.Explain.


10. If CD � 3 and C �D� � 8, is the dilation an enlargement,reduction, or congruence transformation?

W� X�

Y�Z�

WYZX

X �

Y �Z �

W �

a

b

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

R�

P�

Q�

P

QR

NAME DATE PERIOD

SCORE


Chapter 9 Test, Form 2D (continued)99

11. Find the measure of the image of G�H� if GH � 7 under adilation with a scale factor of 5.

12. Draw the image of �CDE under dilation with center G and a scale factor of �

13�.

13. Find the scale factor of the dilation if OP � 15 and O�P� � 20.

14. Find the direction of o� � ��6, �4� to the nearest tenth.

15. Find the magnitude of l�� 5, 12�.

16. Write the component form of EF�.

17. If w� and x� are equal, do they have the same magnitude anddirection?

18. Find the image of the point at (�11, �7) under a translationby r� � ��2, 3�.

19. Use a matrix to find the coordinates of the vertices of theimage of �JKL with J(�5, 4), K(6, 8), and L(�2, �3), underthe translation (x, y) → (x � 6, y � 5).

20. Use a matrix to find the coordinates of the vertices of theimage of �DEF with D(�2, 1), E(�1, 6), and F(3, 2), after areflection in the x-axis.

Bonus An airplane is flying at 300 miles per hour due south. Thewind is blowing from due west at 40 miles per hour. Findthe resultant speed and direction of the plane.

x

y

O

E

F

NAME DATE PERIOD

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

C�

D�

E�

C

D

E

G

Chapter 9 Test, Form 399


Ass

essm

ents

1. Draw the lines of symmetry.

2. Graph �TUV with vertices T(3, 3), U(6, �1), and V(�2, 1).Then graph the image of �TUV reflected in the line y � 2.

3. Name two properties that are preserved by a reflection.

4. Draw the translation image of �ABC in lines � and m.

5. Find the preimage of C��D�� with C�(4, 6) and D�(�1, 2) underthe translation (x, y) → (x � 3, y � 7).

6. Identify the order and magnitude of the rotational symmetryin a regular 20-gon.

7. Find the coordinates of the image and the measure of theangle of rotation if �RST with R(5, 3), S(7, 8), and T(10, 1) isreflected in the line y � x and then in the x-axis.

8. Determine whether a regular decagon can tessellate a plane.Explain.


10. Find the measure of the dilation image of P�Q� if PQ � 12 under a dilation with a scale factor of ��

34�.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

�

AB

C

A� B�

C�

m

V�

T�

U�

T

U

V

NAME DATE PERIOD

SCORE


Chapter 9 Test, Form 3 (continued)99

11. Draw the image of this figure under a dilation with center Cand a scale factor of �

13�.

12. An 11-inch by 14-inch picture is being reduced on a printer bya scale factor of 60%. Find the dimensions of the image.

13. If WX � �23� and W�X� � �

45�, find the scale factor of the dilation.

14. Name the series of reflections that would result in the sameimage as a figure rotated 180° counterclockwise about the origin.

15. Find the direction of p� � �6, �3� to the nearest tenth.

16. Find the magnitude of m�� 7, 2� to the nearest tenth.

17. Write the component form of XY� with X(�11, �3), and Y(�8, 5).

18. Find the image of the point at (4, �7) under the translation byw�� a, b�.

19. Use matrices to find the coordinates of the image of �JKLwith J(�6, �2), K(2, 10), and L(�2, �2), under a dilation with

a scale factor of ��12�, then a reflection in the line y � x, then

the translation (x, y) → (x � 3, y � 2).

20. Use a matrix to find the coordinates of the vertices of theimage of �GHI with G(�5, �4), H(1, 1), and I(6, �2), after areflection in the origin.

Bonus D(0, 6) is rotated 30° clockwise about the origin. Find thecoordinates of its image.

NAME DATE PERIOD

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

C

Chapter 9 Open-Ended Assessment99


Ass

essm

ents

Demonstrate your knowledge by giving a clear, concise solution toeach problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate beyond the requirements of the problem.

1. Draw a uniform regular tessellation. Explain why your figure fits thiscategory.

2. Name an object that has at least one line of symmetry. Describe the linesof symmetry in this object.

3. a. Show the matrix used to determine the vertices of a figure’s image aftera rotation of 90° counterclockwise about the origin.

b. Find the coordinates of the image of B(1, 0) rotated 90°counterclockwise about the origin.

c. Find the coordinates of the image of A(0, 1) rotated 90°counterclockwise about the origin.

d. Compare the answers in parts b and c to the answer in part a.

4. Draw a tessellation that is not uniform.

5. Graph �ABC and label the coordinates of its vertices. Find the image of�ABC under a composition of a reflection, rotation, translation, anddilation. Name the transformations you used.

6. Draw CD� with one endpoint at the origin and the other in the firstquadrant. Find the magnitude and direction of CD� to the nearest tenth.

NAME DATE PERIOD

SCORE


Chapter 9 Vocabulary Test/Review99

Choose from the terms above to complete each sentence.

1. A congruence transformation is called a(n) .

2. A transformation representing a flip of a figure is called a(n).

3. A(n) is a transformation that turns every point of apreimage through a specified angle and direction about a fixedpoint.

4. A vector is in when the initial point is at the origin.

5. A(n) is a transformation that moves all points of afigure the same distance in the same direction.

6. A(n) is a directed segment representing a quantity.

7. When a figure can be folded so that the two halves matchexactly, the fold is called a(n) .

8. A(n) is a transformation that requires a center pointand a scale factor.

9. Two vectors are if and only if they have the same oropposite directions.

10. Expressing a vector as an ordered pair is the of thevector.

Define each term.

11. center of rotation

12. scalar

13. tessellation

?

?

?

?

?

?

?

?

?

? 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

angle of rotationcenter of rotationcolumn matrixcomponent form composition of reflectionscongruence transformationdilationdirect isometrydirectionequal vectors

glide reflectionindirect isometryisometryline of reflectionline of symmetrymagnitudeparallel vectorspoint of symmetryreflection

reflection matrixregular tessellationresultantrotationrotation matrixrotational symmetryscalarscalar multiplicationsemi-regular tessellation

similarity transformationstandard positiontessellationtransformationtranslationtranslation matrixuniformvectorvertex matrix

NAME DATE PERIOD

SCORE

Chapter 9 Quiz (Lessons 9–1 and 9–2)

99


Ass

essm

ents

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

1. Name the coordinates of the image of Q(6, �4) reflected in thex-axis.

2. How many lines of symmetry does an isosceles triangle have?

3. Why is �A�B�C� with vertices A�(�1, �2), B�(0, 0), and C�(�6, 0) not a translation image of �ABC with A(1, 2),B(0, 0), and C(6, 0)?

4. Complete this statement. The image of A(�3, �5) under (x, y) → is A�(6, �1).

5. Draw the image of �PQR with P(0, 4), Q(2, 8), and R(�3, 6)reflected in the line x � 1.

?

Q�

P�

R�


99

1.

2.

3.

4.

5.

1. Find the image of A(�2, 3) reflected in the line y � �x andthen in the x-axis.

2. Draw the image of �PQR under a rotation 45° clockwise aboutpoint T.

3. Determine whether a scalene triangle will tessellate the plane.

4. Determine whether a regular octagon and a square, bothhaving sides 2 units long, can be used to draw a semi-regulartessellation.

5. A figure is reflected in two intersecting lines forming arotation of magnitude 84°. Find the measure of the acuteangle formed by the intersecting lines of reflection.

P�Q�

R�R

Q

P

T

NAME DATE PERIOD

SCORE



99

1.

2.

3.

4.

5.

1. Name a property that is not preserved by a dilation.

2. �XYZ has vertices X(�1, 3), Y(5, 7), and Z(2, �4). Find thecoordinates of the image of �XYZ after a dilation centered atthe origin with a scale factor of 4.

3. Find the magnitude and direction to the nearest degree of x� � �1, �4�.

4. Graph the image of �DEF with vertices D(�1, 2), E(1, 0), andF(�3, �2) under a translation by w�� 0, 2�.

5. STANDARDIZED TEST PRACTICE Reva wants to enlarge a3-inch by 4-inch picture by a scale factor of 2.5. Find thedimensions of the enlarged image.A. 6 in. by 8 in. B. 7.5 in. by 10 in.C. 8 in. by 10 in. D. 9 in. by 12 in.

D�

F�

E�D

E

F

NAME DATE PERIOD

SCORE

Chapter 9 Quiz (Lesson 9–7)

99

1.

2.

3.

4.

5.

1. Use scalar multiplication to dilate Y�Z� with Z(�2, 3) and Y(�1, 4) so that its length is 4 times the length of Y�Z�.

2. Use a matrix to find the coordinates of the vertices of theimage of �MNO with M(�5, 4), N(�3, 3), and O(�4, 7) underthe translation (x, y) → (x � 2, y � 1).

3. Use a matrix to find the coordinates of the vertices of theimage of �ABC with A(�3, �1), B(8, 2), and C(5, 7) after areflection in the y-axis.

4. Find the coordinates of the image of �DEF with D(2, 3),E(4, �1), and F(�2, �3) after a dilation with a scale factor of�12� and then a reflection in the x-axis.

5. Name the matrix by which you would multiply a vertex matrixof a figure on the left to find the figure’s image under arotation 90° counterclockwise about the origin.

NAME DATE PERIOD

SCORE

Chapter 9 Mid-Chapter Test (Lessons 9–1 through 9–4)

99


Ass

essm

ents

1. How many lines of symmetry does a parallelogram have?A. 0 B. 1 C. 2 D. 4

2. Which translation moves every point of a preimage 4 units left and 6 units up?A. (x, y) → (x � 4, y � 6) B. (x, y) → (x � 4, y � 6)C. (x, y) → (x � 6, y � 4) D. (x, y) → (x � 6, y � 4)

3. Find the magnitude of the rotational symmetry in a square.A. 45° B. 90° C. 180° D. 360°

4. Which pair of figures could be used to create a semi-regular tessellation?A. trapezoid, square B. kite, squareC. equilateral triangle, square D. rectangle, square

5. Which action represents the reflection of a figure?A. slide B. shift C. turn D. flip

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

Part II

6. Write the coordinates of the image of S(�7, 1) reflected in they-axis.

7. DEFG is a square with D(1, 1), E(1, 6), F(6, 6), and G(6, 1) andis first reflected in the line x � 1 and then in the y-axis. Findthe coordinates of D�, E �, F �, and G�.

8. Determine whether �P�Q�R� is a translation image of �PQR. Explain.

9. Find the image of B(4, 7) reflected in the y-axis and then inthe line y � �x.

10. Determine whether a regular 30-gon will tessellate the plane.Explain.

P�

Q�

R�

R

Q

P

Part I Write the letter for the correct answer in the blank at the right of each question.


Chapter 9 Cumulative Review(Chapters 1–9)

99

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

1. If �T and �S form a linear pair and m�T is twelve more thaneleven times m�S, find m�T and m�S. (Lesson 1-5)

2. Determine whether the statement If C�D� � D�E�, then D is themidpoint of C�E� is always, sometimes, or never true. (Lesson 2-5)

3. Find the distance between the parallel lines � and m whoseequations are y � x � 6 and y � x � 2, respectively. (Lesson 3-6)

4. Name the missing coordinates of isosceles �DFG with a base n units long. (Lesson 4-7)

5. Two sides of a triangle measure 45 inches and 68 inches, andthe length of the third side measures x inches. Find the rangefor x. (Lesson 5-4)

6. Find x so that W�V� || Y�Z�. (Lesson 6-4) 7. Find b. (Lesson 7-1)

8. In �STV, s � 40, t � 52, and m�T � 82. Find m�S, m�V,and v to the nearest tenth. (Lesson 7-7)

9. Determine whether ABCD is a rectangle with vertices A(2, 8),B(0, 1), C(1, �8), and D(5, 6). Justify your answer. (Lesson 8-4)

10. Find a, b, and m�HJK if HJKL is a parallelogram. (Lesson 8-5)

11. Draw the image of �PQR under a 90° counterclockwiserotation about M(�1, 2). (Lesson 9-3)

12. Write the component form of AB� with A(�4, 7) and B(9, �2).(Lesson 9-6)

(5a � 1)�

(91 � 4a)�

5b � 4 4b

H J

L K

9 5

a cb

21

6xX ZV

YW28

x

y F(?, ?)

G(n, 0)

b

D

P�

Q�

R�

x

y

O

P

QR

M

NAME DATE PERIOD

SCORE

Standardized Test Practice (Chapters 1–9)


1. Which statement has a false converse? (Lesson 2-3)

A. If a quadrilateral is a rectangle, then the diagonals of thequadrilateral are congruent.

B. If a quadrilateral has diagonals that bisect each other, then thequadrilateral is a parallelogram.

C. If a quadrilateral is a rectangle, then all angles of thequadrilateral are right angles.

D. If a quadrilateral is a parallelogram, then both pairs of oppositesides of the quadrilateral are congruent.

2. If �FGH is an isosceles triangle with �G as the vertex angle andm�G � 52, find m�F and m�H. (Lesson 4-6)

E. m�F � 72, m�H � 56 F. m�F � 64, m�H � 64G. m�F � 52, m�H � 76 H. m�F � 40, m�H � 88

3. What can you conclude from the figure? (Lesson 5-5)

A. m�W � m�D by SSSInequalityB. m�G � m�V by SSS InequalityC. DF � WV by SAS InequalityD. GF � VT by SAS Inequality

4. Find x. (Lesson 6-3)

E. 8.68 F. 20.25G. 31.24 H. 42.31

5. If Q�R� is the hypotenuse of right �PQR,PQ � 18, and QR � 24, find RP. (Lesson 7-2)

A. �63� B. �252� C. 30 D. 60

6. If A�B� is a median of trapezoid WXYZ, find m�1, m�2, and m�3. (Lesson 8-6)

E. m�1 � 72, m�2 � 108, m�3 � 72F. m�1 � 108, m�2 � 72, m�3 � 72G. m�1 � 108, m�2 � 72, m�3 � 108H. m�1 � 108, m�2 � 108, m�3 � 72

7. If A(c, d) is reflected in the y-axis, find the coordinates of A�.(Lesson 9-1)

A. A�(c, �d) B. A�(�c, d) C. A�(�c, �d) D. A�(d, c)

W X

BA

Z Y72�123

12.2

13.5

18.325.4�

25.4�

x

22

2014

149

9

W T

V

G F

D

NAME DATE PERIOD

SCORE 99

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.

1.

2.

3.

4.

5.

6.

7. A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

Ass

essm

ents


Standardized Test Practice (continued)

8. Find x so that the line containing (�7, 8) and(6, x) is perpendicular to RT�� with R(3, 9) andT(4, �4). (Lesson 3-3)

9. Find x. (Lesson 4-2)

10. Quadrilateral QRTV � quadrilateral HJKL,QR � 5, RT � 10, JK � 14, and KL � 8. Findthe scale factor as a fraction of quadrilateralQRTV to quadrilateral HJKL. (Lesson 6-2)

11. Find cos M. (Lesson 7-4)

12. Find the measure of the preimage of dilated J�K� if J�K� � 62 and r � ��

13�. (Lesson 9-5)

12

159

K

L M

97.8�

22.1�x �

NAME DATE PERIOD

99

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

Part 3: Short Response

Instructions: Show your work or explain in words how you found your answer.

8. 9.

10. 11.

12.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

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.

99 9 987654321

87654321

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87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

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87654321

0 0 0

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.

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0 0 0

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13. Two parallel lines are cut by a transversal, �1 and �2 areconsecutive interior angles, m�1 � 6y � 5, and m�2 � 10y � 17. Find m�1 and m�2. (Lesson 3-2)

14. Parallelogram ABCD has vertices A(c, a), C(b, 0), and D(0, 0).Find the coordinates for vertex B. (Lesson 8-7)

15. Triangle DEF with D(7, �12), E(2, 10), and F(�11, �8) istranslated so that D�(12, �16). Find the coordinates of E�and F �. (Lesson 9-2)

13.

14.

15.

9 6 0 . 1

5 / 7

1 8 6

0 . 8

Standardized Test PracticeStudent Record Sheet (Use with pages 518–519 of the Student Edition.)

99

© Glencoe/McGraw-Hill A1 Glencoe Geometry

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7

2 5

3 6 DCBADCBA

DCBADCBA

DCBADCBADCBA

NAME DATE PERIOD

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 3 Open-EndedPart 3 Open-Ended

Solve the problem and write your answer in the blank.

For Questions 10 and 11, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.

8 9 10

9 (grid in)

10 (grid in)0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

Record your answers for Questions 11–12 on the back of this paper.


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fig

ure

in

a

poin

t,a

lin

e,or

a p

lan

e.T

he

new

fig

ure

is

the

imag

ean

d th

e or

igin

al f

igu

re i

s th

ep

reim

age.

Th

e pr

eim

age

and

imag

e ar

e co

ngr

uen

t,so

a r

efle

ctio

n i

s a

con

gru

ence

tran

sfor

mat

ion

or i

som

etry

.

Con

stru

ct t

he

imag

eof

qu

adri

late

ral

AB

CD

un

der

are

flec

tion

in

lin

em

.

Dra

w a

per

pend

icul

ar f

rom

eac

h ve

rtex

of t

he q

uadr

ilat

eral

to

m.F

ind

vert

ices

A�,

B�,

C�,

and

D�

that

are

the

sam

edi

stan

ce f

rom

mon

the

oth

er s

ide

of m

.T

he i

mag

e is

A�B

�C�D

�.

m

CA

D

B

A�

B�

C�

D�

Qu

adri

late

ral

DE

FG

has

vert

ices

D(�

2,3)

,E(4

,4),

F(3

,�2)

,an

d

G(�

3,�

1).F

ind

th

e im

age

un

der

ref

lect

ion

in t

he

x-ax

is.

To

fin

d an

im

age

for

are

flec

tion

in

th

e x-

axis

,us

e th

e sa

me

x-co

ordi

nate

and

mu

ltip

ly t

he

y-co

ordi

nat

e by

�1.

Insy

mbo

ls,(

a,b)

→(a

,�b)

.T

he

new

coo

rdin

ates

are

D�(

�2,

�3)

,E�(

4,�

4),

F�(

3,2)

,an

d G

�(�

3,1)

.T

he

imag

e is

D�E

�F�G

�.

x

y

O

D�

E�

F�

G�D

E

FG

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

In E

xam

ple

2,th

e n

otat

ion

(a,

b) →

(a,�

b) r

epre

sen

ts a

ref

lect

ion

in

th

e x-

axis

.Her

e ar

eth

ree

oth

er c

omm

on r

efle

ctio

ns

in t

he

coor

din

ate

plan

e.•

in t

he

y-ax

is:

(a,b

) →

(�a,

b)•

in t

he

lin

e y

�x:

(a,b

) →

(b,a

)•

in t

he

orig

in:

(a,b

) →

(�a,

�b)

Dra

w t

he

imag

e of

eac

h f

igu

re u

nd

er a

ref

lect

ion

in

lin

em

.

1.2.

3.

Gra

ph

eac

h f

igu

re a

nd

its

im

age

un

der

th

e gi

ven

ref

lect

ion

.

4.�

DE

Fw

ith

D(�

2,�

1),E

(�1,

3),

5.A

BC

Dw

ith

A(1

,4),

B(3

,2),

C(2

,�2)

,F

(3,�

1) i

n t

he

x-ax

isD

(�3,

1) i

n t

he

y-ax

is x

y

OD

�

C�

B�

A�

D

C

B

A

x

y

O

F�

D� E

�

FD

E

m

R�

T�

S�

Q

RS

TU

m

P�

N�

M�

LM

N

OP

mH

�

J�K

�

HJ

K

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill48

0G

lenc

oe G

eom

etry

Lin

es a

nd

Po

ints

of

Sym

met

ryIf

a f

igu

re h

as a

lin

e of

sym

met

ry,t

hen

it

can

be

fold

ed a

lon

g th

at l

ine

so t

hat

th

e tw

o h

alve

s m

atch

.If

a fi

gure

has

a p

oin

t of

sym

met

ry,i

tis

th

e m

idpo

int

of a

ll s

egm

ents

bet

wee

n t

he

prei

mag

e an

d im

age

poin

ts.

Det

erm

ine

how

man

y li

nes

of

sym

met

ry

a re

gula

r h

exag

on h

as.T

hen

det

erm

ine

wh

eth

er a

re

gula

r h

exag

on h

as p

oin

t sy

mm

etry

.T

her

e ar

e si

x li

nes

of

sym

met

ry,t

hre

e th

at a

re d

iago

nal

s th

rou

gh o

ppos

ite

vert

ices

an

d th

ree

that

are

per

pen

dicu

lar

bise

ctor

s of

opp

osit

e si

des.

Th

e h

exag

on h

as p

oin

t sy

mm

etry

beca

use

an

y li

ne

thro

ugh

Pid

enti

fies

tw

o po

ints

on

th

e h

exag

on t

hat

can

be

con

side

red

imag

es o

f ea

ch o

ther

.

Det

erm

ine

how

man

y li

nes

of

sym

met

ry e

ach

fig

ure

has

.Th

en d

eter

min

e w

het

her

the

figu

re h

as p

oin

t sy

mm

etry

.

1.2.

3.

4;ye

s3;

no

2;ye

s

4.5.

6.

5;n

o2;

yes

0;n

o

7.8.

9.

1;n

o1;

no

1;n

o

AB

C

DE

FP

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Ref

lect

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-1

9-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises

Answers (Lesson 9-1)


An

swer

s

Skil

ls P

ract

ice

Ref

lect

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-1

9-1

©G

lenc

oe/M

cGra

w-H

ill48

1G

lenc

oe G

eom

etry

Lesson 9-1

Dra

w t

he

imag

e of

eac

h f

igu

re u

nd

er a

ref

lect

ion

in

lin

e �.

1.2.

CO

OR

DIN

ATE

GEO

MET

RYG

rap

h e

ach

fig

ure

an

d i

ts i

mag

e u

nd

er t

he

give

nre

flec

tion

.

3.�

AB

Cw

ith

ver

tice

s A

(�3,

2),B

(0,1

),4.

trap

ezoi

d D

EF

Gw

ith

ver

tice

s D

(0,�

3),

and

C(�

2,�

3) i

n t

he

orig

inE

(1,3

),F

(3,3

),an

d G

(4,�

3) i

n t

he

y-ax

is

5.pa

rall

elog

ram

RS

TU

wit

h v

erti

ces

6.sq

uar

e K

LM

Nw

ith

ver

tice

s K

(�1,

0),

R(�

2,3)

,S(2

,4),

T(2

,�3)

an

d L

(�2,

3),M

(1,4

),an

d N

(2,1

) in

U

(�2,

�4)

in

th

e li

ne

y�

xth

e x-

axis

Det

erm

ine

how

man

y li

nes

of

sym

met

ry e

ach

fig

ure

has

.Th

en d

eter

min

e w

het

her

the

figu

re h

as p

oin

t sy

mm

etry

.

7.8.

9.

1;n

o4;

yes

0;ye

s

M�

K�

Kx

y

O

L�

N�

LM

Nx

y

O

R�S

�T

�

U�

RS T

U

x

y

O

DEF

GD

�

E�

F�

G�

x

y

O

A�

B�

C�

AB

C

�

�

©G

lenc

oe/M

cGra

w-H

ill48

2G

lenc

oe G

eom

etry

Dra

w t

he

imag

e of

eac

h f

igu

re u

nd

er a

ref

lect

ion

in

lin

e �.

1.2.

CO

OR

DIN

ATE

GEO

MET

RYG

rap

h e

ach

fig

ure

an

d i

ts i

mag

e u

nd

er t

he

give

nre

flec

tion

.

3.qu

adri

late

ral

AB

CD

wit

h v

erti

ces

4.�

FG

Hw

ith

ver

tice

s F

(�3,

�1)

,G(0

,4),

A(�

3,3)

,B(1

,4),

C(4

,0),

and

and

H(3

,�1)

in

th

e li

ne

y�

xD

(�3,

�3)

in

th

e or

igin

5.re

ctan

gle

QR

ST

wit

h v

erti

ces

Q(�

3,2)

,6.

trap

ezoi

d H

IJK

wit

h v

erti

ces

H(�

2,5)

,R

(�1,

4),S

(2,1

),an

d T

(0,�

1)

I(2,

5),J

(�4,

�1)

,an

d K

(�4,

3)

in t

he

x-ax

isin

th

e y-

axis

RO

AD

SIG

NS

Det

erm

ine

how

man

y li

nes

of

sym

met

ry e

ach

sig

n h

as.T

hen

det

erm

ine

wh

eth

er t

he

sign

has

poi

nt

sym

met

ry.

7.8.

9.

0;ye

s1;

no

4;ye

s

HI

JK

H�

I�

J�K� x

y

OQ

�

R�

S�

T�

Q

R

S

Tx

y

O

F�

G�

H�

F

G

Hx

y

O

A�

B�

C�

D�

AB

C

D

x

y

O

��Pra

ctic

e (

Ave

rag

e)

Ref

lect

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-1

9-1



Readin

g t

o L

earn

Math

em

ati

csR

efle

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-1

9-1

©G

lenc

oe/M

cGra

w-H

ill48

3G

lenc

oe G

eom

etry

Lesson 9-1

Pre-

Act

ivit

yW

her

e ar

e re

flec

tion

s fo

un

d i

n n

atu

re?

Rea

d th

e in

trod

uct

ion

to

Les

son

9-1

at

the

top

of p

age

463

in y

our

text

book

.

Sup

pose

you

dra

w a

line

seg

men

t co

nnec

ting

a p

oint

at

the

peak

of a

mou

ntai

nto

its

imag

e in

the

lake

.Whe

re w

ill t

he m

idpo

int

of t

his

segm

ent

fall?

on

th

eb

ou

nd

ary

line

bet

wee

n t

he

sho

re a

nd

th

e su

rfac

e o

f th

e la

ke

Rea

din

g t

he

Less

on

1.D

raw

th

e re

flec

ted

imag

e fo

r ea

ch r

efle

ctio

n d

escr

ibed

bel

ow.

a.re

flec

tion

of

trap

ezoi

d A

BC

Din

th

e li

ne

nb

.re

flec

tion

of

�R

ST

in p

oin

t P

Lab

el t

he

imag

e of

AB

CD

as A

�B�C

�D�.

Lab

el t

he

imag

e of

RS

Tas

R�S

�T�.

c.re

flec

tion

of

pen

tago

n A

BC

DE

in t

he

orig

inL

abel

th

e im

age

of A

BC

DE

as A

�B�C

�D�E

�.

2.D

eter

min

e th

e im

age

of t

he

give

n p

oin

t u

nde

r th

e in

dica

ted

refl

ecti

on.

a.(4

,6);

refl

ecti

on i

n t

he

y-ax

is(�

4,6)

b.

(�3,

5);r

efle

ctio

n i

n t

he

x-ax

is(�

3,�

5)c.

(�8,

�2)

;ref

lect

ion

in

th

e li

ne

y�

x(�

2,�

8)d

.(9

,�3)

;ref

lect

ion

in

th

e or

igin

(�9,

3)

3.D

eter

min

e th

e n

um

ber

of l

ines

of

sym

met

ry f

or e

ach

fig

ure

des

crib

ed b

elow

.Th

ende

term

ine

wh

eth

er t

he

figu

re h

as p

oin

t sy

mm

etry

an

d in

dica

te t

his

by

wri

tin

g ye

sor

no.

a.a

squ

are

4;ye

sb

.an

iso

scel

es t

rian

gle

(not

equ

ilat

eral

) 1;

no

c.a

regu

lar

hex

agon

6;

yes

d.a

n i

sosc

eles

tra

pezo

id 1

;n

oe.

a re

ctan

gle

(not

a s

quar

e) 2

;ye

sf.

the

lett

er E

1;

no

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r a

new

geo

met

ric

term

is

to r

elat

e th

e w

ord

or i

ts p

arts

to

geom

etri

c te

rms

you

alr

eady

kn

ow.L

ook

up

the

orig

ins

of t

he

two

part

s of

th

e w

ord

isom

etry

in y

our

dict

ion

ary.

Exp

lain

th

e m

ean

ing

of e

ach

par

t an

d gi

ve a

ter

m y

oual

read

y kn

ow t

hat

sh

ares

th

e or

igin

of

that

par

t.S

amp

le a

nsw

er:T

he

firs

t p

art

com

es f

rom

iso

s,w

hic

h m

ean

s eq

ual

,as

in is

osc

eles

.Th

e se

con

d p

art

com

es f

rom

met

ron

,wh

ich

mea

ns

mea

sure

,as

in g

eom

etry

.

A�

B�

C�

D�

E�

AB

CD

E

x

y

O

R�

S�

T�

RS

P

Tn

A�

B�

C�

D�

A

BC D

©G

lenc

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w-H

ill48

4G

lenc

oe G

eom

etry

Ref

lect

ion

s in

th

e C

oo

rdin

ate

Pla

ne

Stu

dy

the

dia

gram

at

the

righ

t.It

sh

ows

how

th

e tr

ian

gle

AB

Cis

map

ped

on

to t

rian

gle

XY

Zb

y th

etr

ansf

orm

atio

n (

x,y)

→(�

x�

6,y)

.Not

ice

that

�X

YZ

is t

he

refl

ecti

on i

mag

e w

ith

res

pec

t to

th

e ve

rtic

al

lin

e w

ith

eq

uat

ion

x�

3.

1.P

rove

th

at t

he

vert

ical

lin

e w

ith

equ

atio

n x

�3

is t

he

perp

endi

cula

r bi

sect

or o

f th

e se

gmen

t w

ith

en

dpoi

nts

(x

,y)

and

(�x

�6,

y).(

Hin

t:U

se t

he

mid

poin

t fo

rmu

la.)

Mid

po

int

��x

�(�

2x�

6)�

,�y

� 2y

��o

r (3

,y)

Th

e se

gm

ent

join

ing

(x,

y)

and

(�

x�

6,y

) is

h

ori

zon

tal a

nd

hen

ce is

per

pen

dic

ula

r to

th

e ve

rtic

al li

ne.

2.E

very

tra

nsf

orm

atio

n o

f th

e fo

rm (

x,y)

→(�

x�

2h

,y)

is

a re

flec

tion

wit

h r

espe

ct t

o th

e ve

rtic

al l

ine

wit

h e

quat

ion

x

�h

.Wh

at k

ind

of t

ran

sfor

mat

ion

is

(x,y

) →

(x,�

y�

2k)

?

refl

ecti

on

wit

h r

esp

ect

to t

he

ho

rizo

nta

l lin

e w

ith

eq

uat

ion

y�

k

Dra

w t

he

tran

sfor

mat

ion

im

age

for

each

fig

ure

an

d t

he

give

n

tran

sfor

mat

ion

.Is

it a

ref

lect

ion

tra

nsf

orm

atio

n?

If s

o,w

ith

re

spec

t to

wh

at l

ine?

3.(x

,y)

→(�

x�

4,y)

4.(x

,y)

→(x

,�y

�8)

yes;

x�

�2

yes;

y�

4

x

y

Ox

y

O

x �

3

A

B

C

X

Y

Z

(x, y

) → (�

x �

6, y

)

x

y

O

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-1

9-1



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Tran

slat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-2

9-2

©G

lenc

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cGra

w-H

ill48

5G

lenc

oe G

eom

etry

Lesson 9-2

Tran

slat

ion

s U

sin

g C

oo

rdin

ates

A t

ran

sfor

mat

ion

cal

led

a tr

ansl

atio

nsl

ides

afi

gure

in

a g

iven

dir

ecti

on.I

n t

he

coor

din

ate

plan

e,a

tran

slat

ion

mov

es e

very

pre

imag

epo

int

P(x

,y)

to a

n i

mag

e po

int

P(x

�a,

y�

b) f

or f

ixed

val

ues

aan

d b.

In w

ords

,atr

ansl

atio

n s

hif

ts a

fig

ure

au

nit

s h

oriz

onta

lly

and

bu

nit

s ve

rtic

ally

;in

sym

bols

,(x

,y)

→(x

�a,

y�

b).

Rec

tan

gle

RE

CT

has

ver

tice

s R

(�2,

�1)

,E

(�2,

2),C

(3,2

),an

d T

(3,�

1).G

rap

h R

EC

Tan

d i

ts i

mag

e fo

r th

e tr

ansl

atio

n (

x,y)

→(x

�2,

y�

1).

Th

e tr

ansl

atio

n m

oves

eve

ry p

oin

t of

th

e pr

eim

age

righ

t 2

un

its

and

dow

n 1

un

it.

(x,y

) →

(x�

2,y

�1)

R(�

2,�

1) →

R�(

�2

�2,

�1

�1)

or

R�(

0,�

2)E

(�2,

2) →

E�(

�2

�2,

2 �

1) o

r E

�(0,

1)C

(3,2

) →

C�(

3 �

2,2

�1)

or

C�(

5,1)

T(3

,�1)

→T

�(3

�2,

�1

�1)

or

T�(

5,�

2)

Gra

ph

eac

h f

igu

re a

nd

its

im

age

un

der

th

e gi

ven

tra

nsl

atio

n.

1.P �

Q�w

ith

en

dpoi

nts

P(�

1,3)

an

d Q

(2,2

) u

nde

r th

e tr

ansl

atio

n

left

2 u

nit

s an

d u

p 1

un

it

2.�

PQ

Rw

ith

ver

tice

s P

(�2,

�4)

,Q(�

1,2)

,an

d R

(2,1

) u

nde

r th

e tr

ansl

atio

n r

igh

t 2

un

its

and

dow

n 2

un

its

3.sq

uar

e S

QU

Rw

ith

ver

tice

s S

(0,2

),Q

(3,1

),U

(2,�

2),a

nd

R(�

1,�

1) u

nde

r th

e tr

ansl

atio

n r

igh

t 3

un

its

and

up

1 u

nit

x

y

R�S

�Q

�

U�

R

SQ

U

x

y P�

Q�

R�

P

QR

x

yP

�Q

�P

Q

x

y

OE�

R�

C�

T�

E R

C T

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill48

6G

lenc

oe G

eom

etry

Tran

slat

ion

s b

y R

epea

ted

Ref

lect

ion

sA

not

her

way

to

fin

d th

e im

age

of a

tran

slat

ion

is

to r

efle

ct t

he

figu

re t

wic

e in

par

alle

l li

nes

.Th

is k

ind

of t

ran

slat

ion

is

call

ed a

com

pos

ite

of r

efle

ctio

ns.

In t

he

figu

re, m

|| n.F

ind

th

e tr

ansl

atio

n i

mag

e of

�A

BC

.�

A�B

�C�

is t

he

imag

e of

�A

BC

refl

ecte

d in

lin

e m

.�

A�B

�C�

is t

he

imag

e of

�A�

B�C

�re

flec

ted

in l

ine

n.T

he

fin

al i

mag

e,�

A�B

�C�,

is a

tra

nsl

atio

n o

f �

AB

C.

In e

ach

fig

ure

, m|| n

.Fin

d t

he

tran

slat

ion

im

age

of e

ach

fig

ure

by

refl

ecti

ng

it i

nli

ne

man

d t

hen

in

lin

e n.

1.2.

3.4.

5.6.

m

U

A D

QQ

�

U�

A� D

�

U�

A�

D�

n

m n

U

T

S

R U�

T�

S�

R�

m

n

D�

Q�

AQ

U

D

A�

Q�

D�

U�

T�

mn

P�

PP

�

E�

EE

�N

N�

TT

�

AA

�

mn

A�

AB

C

A�

B�

C�

mn

A�

B�

C�

A

B C

A�

B�

C�

mn

A�

B�

C�

A

B C

A�

B�

C�

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Tran

slat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-2

9-2

Exer

cises

Exer

cises

Exam

ple

Exam

ple



Skil

ls P

ract

ice

Tran

slat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-2

9-2

©G

lenc

oe/M

cGra

w-H

ill48

7G

lenc

oe G

eom

etry

Lesson 9-2

In e

ach

fig

ure

,a|| b

.Det

erm

ine

wh

eth

er f

igu

re 3

is

a tr

ansl

atio

n i

mag

e of

fig

ure

1.

Wri

te y

esor

no.

Exp

lain

you

r an

swer

.

1.2.

Yes;

refl

ect

fig

ure

1 in

lin

e a

toN

o;

it is

ori

ente

d d

iffe

ren

tly

than

g

et f

igu

re 2

an

d f

igu

re 2

in li

ne

bfi

gu

re 1

.to

get

fig

ure

3.

3.4.

Yes;

refl

ect

fig

ure

1 in

lin

e a

to

No

;it

is a

ref

lect

ion

of

a tr

ansl

atio

n

get

fig

ure

2 a

nd

fig

ure

2 in

lin

e b

and

is o

rien

ted

dif

fere

ntl

y th

an

to g

et f

igu

re 3

.fi

gu

re 1

.

CO

OR

DIN

ATE

GEO

MET

RYG

rap

h e

ach

fig

ure

an

d i

ts i

mag

e u

nd

er t

he

give

ntr

ansl

atio

n.

5.�

JK

Lw

ith

ver

tice

s J

(�4,

�4)

,6.

quad

rila

tera

l L

MN

Pw

ith

ver

tice

s L

(4,2

),K

(�2,

�1)

,an

d L

(2,�

4) u

nde

r th

e M

(4,�

1),N

(0,�

1),a

nd

P(1

,4)

un

der

the

tran

slat

ion

(x,

y) →

(x�

2,y

�5)

tran

slat

ion

(x,

y) →

(x�

4,y

�3)

M�

L MN

P

L �

N�P

�

x

y

OJ�

K�

L�

J

K

L

x

y

O

ab

12

3

ab

1

23

a b

1 2 3a

b

12

3

©G

lenc

oe/M

cGra

w-H

ill48

8G

lenc

oe G

eom

etry

In e

ach

fig

ure

,c|| d

.Det

erm

ine

wh

eth

er f

igu

re 3

is

a tr

ansl

atio

n i

mag

e of

fig

ure

1.

Wri

te y

esor

no.

Exp

lain

you

r an

swer

.

1.2.

No

;it

is a

tra

nsl

atio

n o

f a

Yes;

it is

a r

efle

ctio

n o

f a

refl

ecti

on

re

flec

tio

n a

nd

is o

rien

ted

wit

h r

esp

ect

to t

he

par

alle

l lin

es.

dif

fere

ntl

y th

an f

igu

re 1

.

CO

OR

DIN

ATE

GEO

MET

RYG

rap

h e

ach

fig

ure

an

d i

ts i

mag

e u

nd

er t

he

give

ntr

ansl

atio

n.

3.qu

adri

late

ral

TU

WX

wit

h v

erti

ces

4.pe

nta

gon

DE

FG

Hw

ith

ver

tice

s D

(�1,

�2)

,T

(�1,

1),U

(4,2

),W

(1,5

),an

d X

(�1,

3)

E(2

,�1)

,F(5

,�2)

,G(4

,�4)

,H(1

,�4)

u

nde

r th

e tr

ansl

atio

n

un

der

the

tran

slat

ion

(x

,y)

→(x

�2,

y�

4)(x

,y)

→(x

�1,

y�

5)

AN

IMA

TIO

NF

ind

th

e tr

ansl

atio

n t

hat

mov

es t

he

figu

re

on t

he

coor

din

ate

pla

ne.

5.fi

gure

1 →

figu

re 2

(x�

4,y

�2)

6.fi

gure

2 →

figu

re 3

(x�

4,y

�1)

7.fi

gure

3 →

figu

re 4

(x�

3,y

�3)

x

y

O

13

4

2

D�

E�

F�

G�

H�

DE

F

GH

x

y

O

W

T�

U�

W�

X�

TU

X

x

y

O

c d

1 2 3

cd

12

3

Pra

ctic

e (

Ave

rag

e)

Tran

slat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-2

9-2



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csTr

ansl

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-2

9-2

©G

lenc

oe/M

cGra

w-H

ill48

9G

lenc

oe G

eom

etry

Lesson 9-2

Pre-

Act

ivit

yH

ow a

re t

ran

slat

ion

s u

sed

in

a m

arch

ing

ban

d s

how

?

Rea

d th

e in

trod

uct

ion

to

Les

son

9-2

at

the

top

of p

age

470

in y

our

text

book

.

How

do

ban

d di

rect

ors

get

the

mar

chin

g ba

nd

to m

ain

tain

th

e sh

ape

of t

he

figu

re t

hey

ori

gin

ally

for

med

?S

amp

le a

nsw

er:T

he

ban

d p

ract

ices

mar

chin

g in

un

iso

n,w

ith

eve

ryo

ne

mak

ing

iden

tica

l mov

es a

tth

e sa

me

tim

e.R

ead

ing

th

e Le

sso

n1.

Un

derl

ine

the

corr

ect

wor

d or

ph

rase

to

form

a t

rue

stat

emen

t.a.

All

ref

lect

ion

s an

d tr

ansl

atio

ns

are

(opp

osit

es/is

omet

ries

/equ

ival

ent)

.b

.T

he

prei

mag

e an

d im

age

of a

fig

ure

un

der

a re

flec

tion

in

a l

ine

hav

e (t

he

sam

e or

ien

tati

on/o

ppos

ite

orie

nta

tion

s).

c.T

he

prei

mag

e an

d im

age

of a

fig

ure

un

der

a tr

ansl

atio

n h

ave

(th

e sa

me

orie

nta

tion

/opp

osit

e or

ien

tati

ons)

.d

.T

he

resu

lt o

f su

cces

sive

ref

lect

ion

s ov

er t

wo

para

llel

lin

es i

s a

(ref

lect

ion

/rot

atio

n/t

ran

slat

ion

).e.

Col

lin

eari

ty (

is/i

s n

ot)

pres

erve

d by

tra

nsl

atio

ns.

f.T

he

tran

slat

ion

(x,

y) →

(x�

a,y

�b)

sh

ifts

eve

ry p

oin

t a

un

its

(hor

izon

tall

y/ve

rtic

ally

) an

d y

un

its

(hor

izon

tall

y/ve

rtic

ally

).

2.F

ind

the

imag

e of

eac

h p

reim

age

un

der

the

indi

cate

d tr

ansl

atio

n.

a.(x

,y);

5 u

nit

s ri

ght

and

3 u

nit

s u

p(x

�5,

y�

3)b

.(x

,y);

2 u

nit

s le

ft a

nd

4 u

nit

s do

wn

(x�

2,y

�4)

c.(x

,y);

1 u

nit

lef

t an

d 6

un

its

up

(x�

1,y

�6)

d.

(x,y

);7

un

its

righ

t(x

�7,

y)e.

(4,�

3);3

un

its

up

(4,0

)f.

(�5,

6);3

un

its

righ

t an

d 2

un

its

dow

n(�

2,4)

g.(�

7,5)

;7 u

nit

s ri

ght

and

5 u

nit

s do

wn

(0,0

)h

.(�

9,�

2);1

2 u

nit

s ri

ght

and

6 u

nit

s do

wn

(3,�

8)

3.�

RS

Th

as v

erti

ces

R(�

3,3)

,S(0

,�2)

,an

d T

(2,1

).G

raph

�R

ST

and

its

imag

e �

R�S

�T�

unde

r th

e tr

ansl

atio

n (x

,y)

→(x

�3,

y�

2).

Lis

t th

e co

ordi

nat

es o

f th

e ve

rtic

es o

f th

e im

age.

R�(

0,1)

,S�(

3,�

4),T

�(5,

�1)

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al t

erm

is

to r

elat

e it

to

an e

very

day

mea

ning

of t

he

sam

e w

ord.

How

is

the

mea

nin

g of

tra

nsl

atio

nin

geo

met

ry r

elat

ed t

o th

e id

ea o

ftr

ansl

atio

nfr

om o

ne

lan

guag

e to

an

oth

er?

Sam

ple

an

swer

:Wh

en y

ou

tra

nsl

ate

fro

m o

ne

lan

gu

age

to a

no

ther

,yo

u c

arry

ove

rth

e m

ean

ing

fro

m o

ne

lan

gu

age

to a

no

ther

.Wh

en y

ou

tra

nsl

ate

a g

eom

etri

c fig

ure

,yo

u c

arry

over

the

figu

re f

rom

on

e p

osi

tion

to

an

oth

er w

itho

ut

chan

gin

g it

s b

asic

R�

S�

T�

R

S

T

x

y

O

©G

lenc

oe/M

cGra

w-H

ill49

0G

lenc

oe G

eom

etry

Tran

slat

ion

s in

Th

e C

oo

rdin

ate

Pla

ne

You

can

use

alg

ebra

ic d

escr

ipti

ons

of r

efle

ctio

ns

to s

how

th

atth

e co

mp

osit

e of

tw

o re

flec

tion

s w

ith

res

pec

t to

par

alle

l li

nes

is

a tr

ansl

atio

n (

that

is,

a sl

ide)

.

1.S

upp

ose

aan

d b

are

two

diff

eren

t re

al n

um

bers

.Let

San

d T

be t

he

foll

owin

g re

flec

tion

s.

S:(

x,y)

→(�

x�

2a,

y)T

:(x,

y) →

(�x

�2

b,y)

Sis

ref

lect

ion

wit

h r

espe

ct t

o th

e li

ne

wit

h e

quat

ion

x�

a,an

d T

isre

flec

tion

wit

h r

espe

ct t

o th

e li

ne

wit

h e

quat

ion

x�

b.

a.F

ind

an a

lgeb

raic

des

crip

tion

(si

mil

ar t

o th

ose

abov

e fo

r S

and

T)

to d

escr

ibe

the

com

posi

te t

ran

sfor

mat

ion

“S

foll

owed

by

T.”

(x,y

) →

(x�

2 (b

�a)

,y)

b.F

ind

an a

lgeb

raic

des

crip

tion

for

th

e co

mpo

site

tra

nsf

orm

atio

n“T

foll

owed

by

S.”

(x,y

) →

(x�

2 (a

�b

),y

)

2.T

hin

k ab

out

the

resu

lts

you

obt

ain

ed i

n E

xerc

ise

1.W

hat

do

they

tell

you

abo

ut

how

th

e di

stan

ce b

etw

een

tw

o pa

rall

el l

ines

is

rela

ted

to t

he

dist

ance

bet

wee

n a

pre

imag

e an

d im

age

poin

t fo

r a

com

posi

te o

f re

flec

tion

s w

ith

res

pect

to

thes

e li

nes

?

Th

e d

ista

nce

bet

wee

n a

pre

imag

e an

d it

s im

age

po

int

istw

ice

the

dis

tan

ce b

etw

een

th

e p

aral

lel l

ines

.

3.Il

lust

rate

you

r an

swer

s to

Exe

rcis

es 1

an

d 2

wit

h s

ketc

hes

.Use

ase

para

te s

hee

t if

nec

essa

ry.

See

stu

den

ts’w

ork

.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-2

9-2



Stu

dy G

uid

e a

nd I

nte

rven

tion

Ro

tati

on

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-3

9-3

©G

lenc

oe/M

cGra

w-H

ill49

1G

lenc

oe G

eom

etry

Lesson 9-3

Dra

w R

ota

tio

ns

A t

ran

sfor

mat

ion

cal

led

a ro

tati

ontu

rns

a fi

gure

th

rou

gh a

spe

cifi

edan

gle

abou

t a

fixe

d po

int

call

ed t

he

cen

ter

of r

otat

ion

.To

fin

d th

e im

age

of a

rot

atio

n,o

ne

way

is

to u

se a

pro

trac

tor.

An

oth

er w

ay i

s to

ref

lect

a f

igu

re t

wic

e,in

tw

o in

ters

ecti

ng

lin

es.

�A

BC

has

ver

tice

s A

(2,1

),B

(3,4

),an

d

C(5

,1).

Dra

w t

he

imag

e of

�A

BC

un

der

a r

otat

ion

of

90°

cou

nte

rclo

ckw

ise

abou

t th

e or

igin

.•

Fir

st d

raw

�A

BC

.Th

en d

raw

a s

egm

ent

from

O,t

he

orig

in,

to p

oin

t A

.•

Use

a p

rotr

acto

r to

mea

sure

90°

cou

nte

rclo

ckw

ise

wit

h O �

A�as

on

e si

de.

•D

raw

OR

� �� .

•U

se a

com

pass

to

copy

O �A�

onto

OR

�� .

Nam

e th

e se

gmen

t O�

A��.

•R

epea

t w

ith

seg

men

ts f

rom

th

e or

igin

to

poin

ts B

and

C.

Fin

d t

he

imag

e of

�A

BC

un

der

ref

lect

ion

in

lin

es m

and

n.

Fir

st r

efle

ct �

AB

Cin

lin

e m

.Lab

el t

he

imag

e �

A�B

�C�.

Ref

lect

�A�

B�C

�in

lin

e n.

Lab

el t

he

imag

e �

A�B

�C�.

�A

�B�C

�is

a r

otat

ion

of

�A

BC

.Th

e ce

nte

r of

rot

atio

n i

s th

ein

ters

ecti

on o

f li

nes

man

d n.

Th

e an

gle

of r

otat

ion

is

twic

e th

em

easu

re o

f th

e ac

ute

an

gle

form

ed b

y m

and

n.

Dra

w t

he

rota

tion

im

age

of e

ach

fig

ure

90°

in t

he

give

n d

irec

tion

ab

out

the

cen

ter

poi

nt

and

lab

el t

he

coor

din

ates

.

1.P�

Q�w

ith

en

dpoi

nts

P(�

1,�

2)

2.�

PQ

Rw

ith

ver

tice

s P

(�2,

�3)

,Q(2

,�1)

,an

d Q

(1,3

) co

un

terc

lock

wis

e an

d R

(3,2

) cl

ockw

ise

abou

t th

e po

int

T(1

,1)

abou

t th

e or

igin

Fin

d t

he

rota

tion

im

age

of e

ach

fig

ure

by

refl

ecti

ng

it i

n l

ine

man

d t

hen

in

lin

e n

.

3.4.

mn

B�

C�

A�

AB

C

B�

C�

A�

mn

Q�

P�

Q

PQ

�

P�

x

yP

�

Q�

R�

P

Q

RT

x

y

P�

Q�

P

Q

m

n

A� B

�C

�

A�

B�

C�

B

C

A

x

y

O

A�

B�

C�

A

B

C

R

Exer

cises

Exer

cises

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

©G

lenc

oe/M

cGra

w-H

ill49

2G

lenc

oe G

eom

etry

Ro

tati

on

al S

ymm

etry

Wh

en t

he

figu

re a

t th

e ri

ght

is r

otat

ed

abou

t po

int

Pby

120

°or

240

°,th

e im

age

look

s li

ke t

he

prei

mag

e.T

he

figu

re h

as r

otat

ion

al s

ymm

etry

,wh

ich

mea

ns

it c

an b

e ro

tate

d le

ss

than

360

°ab

out

a po

int

and

the

prei

mag

e an

d im

age

appe

ar t

o be

th

e sa

me.

Th

e fi

gure

has

rot

atio

nal

sym

met

ry o

f or

der

3 be

cau

se t

her

e ar

e 3

rota

tion

s le

ss t

han

360

°(0

°,12

0°,2

40°)

th

at p

rodu

ce a

n i

mag

e th

at

is t

he

sam

e as

th

e or

igin

al.T

he

mag

nit

ud

eof

th

e ro

tati

onal

sym

met

ry

for

a fi

gure

is

360

degr

ees

divi

ded

by t

he

orde

r.F

or t

he

figu

re a

bove

,th

e ro

tati

onal

sym

met

ry h

as m

agn

itu

de 1

20 d

egre

es.

Iden

tify

th

e or

der

an

d m

agn

itu

de

of t

he

rota

tion

al

sym

met

ry o

f th

e d

esig

n a

t th

e ri

ght.

Th

e de

sign

has

rot

atio

nal

sym

met

ry a

bou

t th

e ce

nte

r po

int

for

rota

tion

s of

0°,

45°,

90°,

135°

,180

°,22

5°,2

70°,

and

315°

.

Th

ere

are

eigh

t ro

tati

ons

less

th

an 3

60 d

egre

es,s

o th

e or

der

of i

ts

rota

tion

al s

ymm

etry

is

8.T

he

quot

ien

t 36

0 �

8 is

45,

so t

he

mag

nit

ude

of

its

rot

atio

nal

sym

met

ry i

s 45

deg

rees

.

Iden

tify

th

e or

der

an

d m

agn

itu

de

of t

he

rota

tion

al s

ymm

etry

of

each

fig

ure

.

1.a

squ

are

2.a

regu

lar

40-g

on

ord

er:

4;m

agn

itu

de:

90°

ord

er:

40;

mag

nit

ud

e:9°

3.4.

ord

er:

2;m

agn

itu

de:

180°

ord

er:

5;m

agn

itu

de:

72°

5.6.

ord

er:

16;

mag

nit

ud

e:22

.5°

ord

er:

3;m

agn

itu

de:

120°

P

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Ro

tati

on

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-3

9-3

Exer

cises

Exer

cises

Exam

ple

Exam

ple



An

swer

s

Skil

ls P

ract

ice

Ro

tati

on

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-3

9-3

©G

lenc

oe/M

cGra

w-H

ill49

3G

lenc

oe G

eom

etry

Lesson 9-3

Rot

ate

each

fig

ure

ab

out

poi

nt

Ru

nd

er t

he

give

n a

ngl

e of

rot

atio

n a

nd

th

e gi

ven

dir

ecti

on.L

abel

th

e ve

rtic

es o

f th

e ro

tati

on i

mag

e.

1.90

°co

un

terc

lock

wis

e2.

90°

cloc

kwis

e

CO

OR

DIN

ATE

GEO

MET

RYD

raw

th

e ro

tati

on i

mag

e of

eac

h f

igu

re 9

0°in

th

e gi

ven

dir

ecti

on a

bou

t th

e or

igin

an

d l

abel

th

e co

ord

inat

es.

3.�

ST

Ww

ith

ver

tice

s S

(2,�

1),T

(5,1

),4.

�D

EF

wit

h v

erti

ces

D(�

4,3)

,E(1

,2),

and

W(3

,3)

cou

nte

rclo

ckw

ise

and

F(�

3,�

3) c

lock

wis

e

Use

a c

omp

osit

ion

of

refl

ecti

ons

to f

ind

th

e ro

tati

on i

mag

e w

ith

res

pec

t to

lin

es k

and

m.T

hen

fin

d t

he

angl

e of

rot

atio

n f

or e

ach

im

age.

5.6.

120°

clo

ckw

ise

140°

cou

nte

rclo

ckw

ise

k mL�

M�

N� O

�

LM N

Ok m

A�

B�C�A

BC

E�

D�

F�

F

ED

x

y

OS

�

T�

W�

S

T

W

x

y

O

GH

KJ

R

G�

H�

J�K�

Q

PS

R

Q�

S�

P�

©G

lenc

oe/M

cGra

w-H

ill49

4G

lenc

oe G

eom

etry

Rot

ate

each

fig

ure

ab

out

poi

nt

Ru

nd

er t

he

give

n a

ngl

e of

rot

atio

n a

nd

th

e gi

ven

dir

ecti

on.L

abel

th

e ve

rtic

es o

f th

e ro

tati

on i

mag

e.

1.80

°co

un

terc

lock

wis

e2.

100°

cloc

kwis

e

CO

OR

DIN

ATE

GEO

MET

RYD

raw

th

e ro

tati

on i

mag

e of

eac

h f

igu

re 9

0°in

th

e gi

ven

dir

ecti

on a

bou

t th

e ce

nte

r p

oin

t an

d l

abel

th

e co

ord

inat

es.

3.�

RS

Tw

ith

ver

tice

s R

(�3,

3),S

(2,4

),4.

�H

JK

wit

h v

erti

ces

H(3

,1),

J(3

,�3)

,an

d T

(1,2

) cl

ockw

ise

abou

t th

e an

d K

(�3,

�4)

cou

nte

rclo

ckw

ise

abou

t po

int

P(1

,0)

the

poin

t P

(�1,

�1)

Use

a c

omp

osit

ion

of

refl

ecti

ons

to f

ind

th

e ro

tati

on i

mag

e w

ith

res

pec

t to

lin

es p

and

s.T

hen

fin

d t

he

angl

e of

rot

atio

n f

or e

ach

im

age.

5.6.

160°

cou

nte

rclo

ckw

ise

100°

cou

nte

rclo

ckw

ise

7.ST

EAM

BO

ATS

A p

addl

e w

hee

l on

a s

team

boat

is

driv

en b

y a

stea

m e

ngi

ne

and

mov

esfr

om o

ne

padd

le t

o th

e n

ext

to p

rope

l th

e bo

at t

hro

ugh

th

e w

ater

.If

a pa

ddle

wh

eel

con

sist

s of

18

even

ly s

pace

d pa

ddle

s,id

enti

fy t

he

orde

r an

d m

agn

itu

de o

f it

s ro

tati

onal

sym

met

ry.

ord

er 1

8 an

d m

agn

itu

de

20°

ps

P�

R�

S�

T�

P

R

S

T

ps

E�

F�

G�

EF

G

K�

J�H

�

JH

K

x

y

OP

( –1,

–1)

S�

R�

T�

T

S

R

x

y

OP

( 1, 0

)

QU

TS

P

RQ�

S�T

�U

� P�

N

MP

R

M�

N�

P�

Pra

ctic

e (

Ave

rag

e)

Ro

tati

on

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-3

9-3



Readin

g t

o L

earn

Math

em

ati

csR

ota

tio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-3

9-3

©G

lenc

oe/M

cGra

w-H

ill49

5G

lenc

oe G

eom

etry

Lesson 9-3

Pre-

Act

ivit

yH

ow d

o so

me

amu

sem

ent

rid

es i

llu

stra

te r

otat

ion

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

9-3

at

the

top

of p

age

476

in y

our

text

book

.

Wh

at a

re t

wo

way

s th

at e

ach

car

rot

ates

?

Eac

h c

ar s

pin

s ar

ou

nd

its

ow

n c

ente

r an

d e

ach

car

ro

tate

sar

ou

nd

a p

oin

t in

th

e ce

nte

r o

f th

e ci

rcu

lar

trac

k.

Rea

din

g t

he

Less

on

1.L

ist

all

of t

he

foll

owin

g ty

pes

of t

ran

sfor

mat

ion

s th

at s

atis

fy e

ach

des

crip

tion

:ref

lect

ion

,tr

ansl

atio

n,r

otat

ion

.a.

Th

e tr

ansf

orm

atio

n i

s an

iso

met

ry.

refl

ecti

on

,tra

nsl

atio

n,r

ota

tio

nb

.T

he

tran

sfor

mat

ion

pre

serv

es t

he

orie

nta

tion

of

a fi

gure

.tr

ansl

atio

n,r

ota

tio

nc.

Th

e tr

ansf

orm

atio

n i

s th

e co

mpo

site

of

succ

essi

ve r

efle

ctio

ns

over

tw

o in

ters

ecti

ng

lin

es.

rota

tio

nd

.T

he

tran

sfor

mat

ion

is

the

com

posi

te o

f su

cces

sive

ref

lect

ion

s ov

er t

wo

para

llel

li

nes

.tr

ansl

atio

ne.

A s

peci

fic

tran

sfor

mat

ion

is

defi

ned

by

a fi

xed

poin

t an

d a

spec

ifie

d an

gle.

rota

tio

nf.

A s

peci

fic

tran

sfor

mat

ion

is

defi

ned

by

a fi

xed

poin

t,a

fixe

d li

ne,

or a

fix

ed p

lan

e.re

flec

tio

ng.

A s

peci

fic

tran

sfor

mat

ion

is

defi

ned

by

(x,y

) →

(x�

a,x

�b)

,for

fix

ed v

alu

es o

f a

and

b.tr

ansl

atio

nh

.T

he

tran

sfor

mat

ion

is

also

cal

led

a sl

ide.

tran

slat

ion

i.T

he

tran

sfor

mat

ion

is

also

cal

led

a fl

ip.

refl

ecti

on

j.T

he

tran

sfor

mat

ion

is

also

cal

led

a tu

rn.

rota

tio

n

2.D

eter

min

e th

e or

der

and

mag

nit

ude

of

the

rota

tion

al s

ymm

etry

for

eac

h f

igu

re.

a.b

.

ord

er 3

;m

agn

itu

de

120°

ord

er 2

;m

agn

itu

de

180°

c.d

.

ord

er 5

;m

agn

itu

de

72°

ord

er 8

;m

agn

itu

de

45°

Hel

pin

g Y

ou

Rem

emb

er

3.W

hat

is

an e

asy

way

to

rem

embe

r th

e or

der

and

mag

nit

ude

of

the

rota

tion

al s

ymm

etry

of a

reg

ula

r po

lygo

n?

Sam

ple

an

swer

:Th

e o

rder

is t

he

sam

e as

th

e n

um

ber

of

sid

es.T

o f

ind

th

em

agn

itu

de,

div

ide

360

by t

he

nu

mb

er o

f si

des

.

©G

lenc

oe/M

cGra

w-H

ill49

6G

lenc

oe G

eom

etry

Fin

din

g t

he

Cen

ter

of

Ro

tati

on

Su

ppos

e yo

u a

re t

old

that

�X

�Y�Z

�is

th

e ro

tati

on i

mag

e of

�X

YZ

,bu

t yo

u a

re n

ot t

old

wh

ere

the

cen

ter

of r

otat

ion

is

nor

th

e m

easu

re

of t

he

angl

e of

rot

atio

n.C

an y

ou f

ind

them

? Ye

s,yo

u c

an.C

onn

ect

two

pair

s of

cor

resp

ondi

ng

vert

ices

wit

h s

egm

ents

.In

th

e fi

gure

,th

e se

gmen

ts Y

Y�

and

ZZ

�ar

e u

sed.

Dra

w t

he

perp

endi

cula

r bi

sect

ors,

�an

d m

,of

thes

ese

gmen

ts.T

he

poin

t C

wh

ere

�an

d m

inte

rsec

t is

th

e ce

nte

r of

rot

atio

n.

1.H

ow c

an y

ou f

ind

the

mea

sure

of

the

angl

e of

rot

atio

n i

n t

he

figu

re a

bove

?D

raw

�X

CX

�(�

YC

Y�

or

�Z

CZ

�w

ou

ld

also

do

) an

d m

easu

re it

wit

h a

pro

trac

tor.

Loc

ate

the

cen

ter

of r

otat

ion

for

th

e ro

tati

on t

hat

map

s W

XY

Zon

to W

�X�Y

�Z�.

Th

en f

ind

th

e m

easu

re o

f th

e an

gle

of r

otat

ion

.

2.3.

Th

e se

gm

ents

stu

den

ts c

ho

ose

to

bis

ect

may

var

y.S

ee s

tud

ents

’wo

rk.

Z

Z�

W

W�

X�

X

Y

Y�

cen

ter

of

rota

tio

n

ang

le o

fro

tati

on

:ab

ou

t 11

0

Z�

Z

W

W�

X�

X

Y�

Y

cen

ter

of

rota

tio

n

ang

le o

fro

tati

on

:ab

ou

t 10

0

Z�

Z

C

X�

X

Y�

Y

m�

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-3

9-3



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Tess

ella

tio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-4

9-4

©G

lenc

oe/M

cGra

w-H

ill49

7G

lenc

oe G

eom

etry

Lesson 9-4

Reg

ula

r Te

ssel

lati

on

sA

pat

tern

th

at c

over

s a

plan

e w

ith

rep

eati

ng

copi

es o

f on

e or

mor

e fi

gure

s so

th

at t

her

e ar

e n

o ov

erla

ppin

g or

em

pty

spac

es i

s a

tess

ella

tion

.A r

egu

lar

tess

ella

tion

uses

onl

y on

e ty

pe o

f re

gula

r po

lygo

n.In

a t

esse

llat

ion,

the

sum

of

the

mea

sure

sof

the

ang

les

of t

he p

olyg

ons

surr

ound

ing

a ve

rtex

is 3

60.I

f a

regu

lar

poly

gon

has

an

in

teri

oran

gle

that

is

a fa

ctor

of

360,

then

th

e po

lygo

n w

ill

tess

ella

te.

regu

lar

tess

ella

tion

tess

ella

tion

Cop

ies

of a

reg

ular

hex

agon

C

opie

s of

a r

egul

ar p

enta

gon

can

form

a t

esse

llatio

n.ca

nnot

for

m a

tes

sella

tion.

Det

erm

ine

wh

eth

er a

reg

ula

r 16

-gon

tes

sell

ates

th

e p

lan

e.E

xpla

in.

If m

�1

is t

he

mea

sure

of

one

inte

rior

an

gle

of a

reg

ula

r po

lygo

n,t

hen

a f

orm

ula

for

m�

1

is m

�1

��18

0(n n

�2)

�.U

se t

he

form

ula

wit

h n

�16

.

m�

1��18

0(n n

�2)

�

��18

0(1 16 6

�2)

�

�15

7.5

Th

e va

lue

157.

5 is

not

a f

acto

r of

360

,so

the

16-g

on w

ill

not

tes

sell

ate.

Det

erm

ine

wh

eth

er e

ach

pol

ygon

tes

sell

ates

th

e p

lan

e.If

so,

dra

w a

sam

ple

fig

ure

.

1.sc

alen

e ri

ght

tria

ngl

eye

s2.

isos

cele

s tr

apez

oid

yes

Det

erm

ine

wh

eth

er e

ach

reg

ula

r p

olyg

on t

esse

llat

es t

he

pla

ne.

Exp

lain

.

3.sq

uar

e4.

20-g

on

Yes;

the

mea

sure

of

each

inte

rio

r N

o;

the

mea

sure

of

each

inte

rio

r an

gle

is 9

0,an

d 9

0 is

a f

acto

r an

gle

is 1

62,a

nd

162

is n

ot

a fa

cto

r o

f 36

0.o

f 36

0.

5.se

ptag

on6.

15-g

on

No

;th

e m

easu

re o

f ea

ch in

teri

or

No

;th

e m

easu

re o

f ea

ch in

teri

or

ang

le is

128

.6,a

nd

128

.6 is

no

t a

ang

le is

156

,an

d 1

56 is

no

t a

fact

or

fact

or

of

360.

of

360.

7.oc

tago

n8.

pen

tago

n

No

;th

e m

easu

re o

f ea

ch in

teri

or

No

;th

e m

easu

re o

f ea

ch in

teri

or

ang

le is

135

,an

d 1

35 is

no

t a

ang

le is

108

,an

d 1

08 is

no

t a

fact

or

fact

or

of

360.

of

360.

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill49

8G

lenc

oe G

eom

etry

Tess

ella

tio

ns

wit

h S

pec

ific

Att

rib

ute

sA

tes

sell

atio

n p

atte

rn c

an c

onta

in a

ny

type

of p

olyg

on.I

f th

e ar

ran

gem

ent

of s

hap

es a

nd

angl

es a

t ea

ch v

erte

x in

th

e te

ssel

lati

on i

s th

esa

me,

the

tess

ella

tion

is

un

ifor

m.A

sem

i-re

gula

r te

ssel

lati

onis

a u

nif

orm

tes

sell

atio

nth

at c

onta

ins

two

or m

ore

regu

lar

poly

gon

s.

Det

erm

ine

wh

eth

er a

kit

e w

ill

tess

ella

te t

he

pla

ne.

If s

o,d

escr

ibe

the

tess

ella

tion

as

un

ifor

m,r

egu

lar,

sem

i-re

gula

r,or

not

un

ifor

m.

A k

ite

wil

l te

ssel

late

th

e pl

ane.

At

each

ver

tex

the

sum

of

th

e m

easu

res

is a

�b

�b

�c,

wh

ich

is

360.

Th

e te

ssel

lati

on i

s u

nif

orm

.

Det

erm

ine

wh

eth

er a

sem

i-re

gula

r te

ssel

lati

on c

an b

e cr

eate

d f

rom

eac

h s

et o

ffi

gure

s.If

so,

sket

ch t

he

tess

ella

tion

.Ass

um

e th

at e

ach

fig

ure

has

a s

ide

len

gth

of

1 u

nit

.

1.rh

ombu

s,eq

uil

ater

al t

rian

gle,

2.sq

uar

e an

d eq

uil

ater

al t

rian

gle

and

octa

gon

no

yes

Det

erm

ine

wh

eth

er e

ach

pol

ygon

tes

sell

ates

th

e p

lan

e.If

so,

des

crib

e th

ete

ssel

lati

on a

s u

nif

orm

,not

un

ifor

m,r

egu

lar,

or s

emi-

regu

lar.

3.re

ctan

gle

4.h

exag

on a

nd

squ

are

yes,

un

iform

no

c�a�

b� b�

c�c�

a�

a�b�

b� b�

c�a�

b� b�

c�a�

b� b�

c�a�

b� b�

b�

c�a�

b� b�

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Tess

ella

tio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-4

9-4

Exer

cises

Exer

cises

Exam

ple

Exam

ple



Skil

ls P

ract

ice

Tess

ella

tio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-4

9-4

©G

lenc

oe/M

cGra

w-H

ill49

9G

lenc

oe G

eom

etry

Lesson 9-4

Det

erm

ine

wh

eth

er e

ach

reg

ula

r p

olyg

on t

esse

llat

es t

he

pla

ne.

Exp

lain

.

1.15

-gon

2.18

-gon

no

,in

teri

or

ang

le �

156

no

,in

teri

or

ang

le �

160

3.sq

uar

e4.

20-g

onye

s,in

teri

or

ang

le �

90n

o,i

nte

rio

r an

gle

�16

2

Det

erm

ine

wh

eth

er a

sem

i-re

gula

r te

ssel

lati

on c

an b

e cr

eate

d f

rom

eac

h s

et o

ffi

gure

s.A

ssu

me

each

fig

ure

has

a s

ide

len

gth

of

1 u

nit

.

5.re

gula

r pe

nta

gon

s an

d eq

uil

ater

al t

rian

gles

no

6.re

gula

r do

deca

gon

s an

d eq

uil

ater

al t

rian

gles

yes

7.re

gula

r oc

tago

ns

and

equ

ilat

eral

tri

angl

esn

o

Det

erm

ine

wh

eth

er e

ach

pol

ygon

tes

sell

ates

th

e p

lan

e.If

so,

des

crib

e th

ete

ssel

lati

on a

s u

nif

orm

,not

un

ifor

m,r

egu

lar,

or s

emi-

regu

lar.

8.rh

ombu

s9.

isos

cele

s tr

apez

oid

and

squ

are

yes;

un

iform

no

Det

erm

ine

wh

eth

er e

ach

pat

tern

is

a te

ssel

lati

on.I

f so

,des

crib

e it

as

un

ifor

m,n

otu

nif

orm

,reg

ula

r,or

sem

i-re

gula

r.

10.

11.

yes;

un

iform

,sem

i-re

gu

lar

yes;

un

iform

©G

lenc

oe/M

cGra

w-H

ill50

0G

lenc

oe G

eom

etry

Det

erm

ine

wh

eth

er e

ach

reg

ula

r p

olyg

on t

esse

llat

es t

he

pla

ne.

Exp

lain

.

1.22

-gon

2.40

-gon

no

,in

teri

or

ang

le ≈

163.

6n

o,i

nte

rio

r an

gle

�17

1

Det

erm

ine

wh

eth

er a

sem

i-re

gula

r te

ssel

lati

on c

an b

e cr

eate

d f

rom

eac

h s

et o

ffi

gure

s.A

ssu

me

each

fig

ure

has

a s

ide

len

gth

of

1 u

nit

.

3.re

gula

r pe

nta

gon

s an

d re

gula

r de

cago

ns

no

(w

ill n

ot

tess

ella

te t

he

pla

ne)

4.re

gula

r do

deca

gon

s,re

gula

r h

exag

ons,

and

squ

ares

yes

Det

erm

ine

wh

eth

er e

ach

pol

ygon

tes

sell

ates

th

e p

lan

e.If

so,

des

crib

e th

ete

ssel

lati

on a

s u

nif

orm

,not

un

ifor

m,r

egu

lar,

or s

emi-

regu

lar.

5.ki

te6.

octa

gon

an

d de

cago

n

yes;

un

iform

no

Det

erm

ine

wh

eth

er e

ach

pat

tern

is

a te

ssel

lati

on.I

f so

,des

crib

e it

as

un

ifor

m,n

otu

nif

orm

,reg

ula

r,or

sem

i-re

gula

r.

7.8.

yes;

un

iform

,sem

i-re

gu

lar

yes,

no

t u

nifo

rm

FLO

OR

TIL

ESF

or E

xerc

ises

9 a

nd

10,

use

th

e fo

llow

ing

info

rmat

ion

.M

r.M

arti

nez

ch

ose

the

patt

ern

of

tile

sh

own

to

reti

le h

is k

itch

en f

loor

.

9.D

eter

min

e w

het

her

th

e pa

tter

n i

s a

tess

ella

tion

.Exp

lain

Yes;

sin

ce t

her

e ar

e 2

squ

ares

an

d 3

tri

ang

les

at e

ach

ve

rtex

,th

e su

m o

f th

e an

gle

s at

th

e ve

rtic

es is

360

°.

10.I

s th

e pa

tter

n u

nif

orm

,reg

ula

r,or

sem

i-re

gula

r?u

nifo

rm a

nd

sem

i-re

gu

lar

Pra

ctic

e (

Ave

rag

e)

Tess

ella

tio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-4

9-4



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csTe

ssel

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-4

9-4

©G

lenc

oe/M

cGra

w-H

ill50

1G

lenc

oe G

eom

etry

Lesson 9-4

Pre-

Act

ivit

yH

ow a

re t

esse

llat

ion

s u

sed

in

art

?

Rea

d th

e in

trod

uct

ion

to

Les

son

9-4

at

the

top

of p

age

483

in y

our

text

book

.

•In

th

e pa

tter

n s

how

n i

n t

he

pict

ure

in

you

r te

xtbo

ok,h

ow m

any

smal

leq

uil

ater

al t

rian

gles

mak

e u

p on

e re

gula

r h

exag

on?

6•

In t

his

pat

tern

,how

man

y fi

sh m

ake

up

one

equ

ilat

eral

tri

angl

e?1�

1 2�

Rea

din

g t

he

Less

on

1.U

nde

rlin

e th

e co

rrec

t w

ord,

phra

se,o

r n

um

ber

to f

orm

a t

rue

stat

emen

t.

a.A

tes

sell

atio

n i

s a

patt

ern

th

at c

over

s a

plan

e w

ith

th

e sa

me

figu

re o

r se

t of

fig

ure

s so

that

th

ere

are

no

(con

gru

ent

angl

es/o

verl

appi

ng

or e

mpt

y sp

aces

/rig

ht

angl

es).

b.

A t

esse

llat

ion

th

at u

ses

only

on

e ty

pe o

f re

gula

r po

lygo

n i

s ca

lled

a(u

nif

orm

/reg

ula

r/se

mi-

regu

lar)

tes

sell

atio

n.

c.T

he

sum

of

the

mea

sure

s of

th

e an

gles

at

any

vert

ex i

n a

ny

tess

ella

tion

is

(90/

180/

360)

.

d.

A t

esse

llat

ion

th

at c

onta

ins

the

sam

e ar

ran

gem

ent

of s

hap

es a

nd

angl

es a

t ev

ery

vert

ex i

s ca

lled

a (

un

ifor

m/r

egu

lar/

sem

i-re

gula

r) t

esse

llat

ion

.

e.In

a r

egu

lar

tess

ella

tion

mad

e u

p of

hex

agon

s,th

ere

are

(3/4

/6)

hex

agon

s m

eeti

ng

atea

ch v

erte

x,an

d th

e m

easu

re o

f ea

ch o

f th

e an

gles

at

any

vert

ex i

s (6

0/90

/120

).

f.A

un

ifor

m t

esse

llat

ion

for

med

usi

ng

two

or m

ore

regu

lar

poly

gon

s is

cal

led

a(r

otat

ion

al/r

egu

lar/

sem

i-re

gula

r) t

esse

llat

ion

.

g.In

a r

egu

lar

tess

ella

tion

mad

e u

p of

tri

angl

es,t

her

e ar

e (3

/4/6

) tr

ian

gles

mee

tin

g at

each

ver

tex,

and

the

mea

sure

of

each

of

the

angl

es a

t an

y ve

rtex

is

(30/

60/1

20).

h.

If a

reg

ula

r te

ssel

lati

on i

s m

ade

up

of q

uad

rila

tera

ls,a

ll o

f th

e qu

adri

late

rals

mu

st b

eco

ngr

uen

t (r

ecta

ngl

es/p

aral

lelo

gram

s/sq

uar

es/t

rape

zoid

s).

2.W

rite

all

of

the

foll

owin

g w

ords

th

at d

escr

ibe

each

tes

sell

atio

n:u

nif

orm

,non

-un

ifor

m,

regu

lar,

sem

i-re

gula

r.

a.n

on

-un

iform

b.

un

iform

,sem

i-re

gu

lar

Hel

pin

g Y

ou

Rem

emb

er

3.O

ften

th

e ev

eryd

ay m

ean

ings

of

a w

ord

can

hel

p yo

u t

o re

mem

ber

its

mat

hem

atic

alm

ean

ing.

Loo

k u

p u

nif

orm

in y

our

dict

ion

ary.

How

can

its

eve

ryda

y m

ean

ings

hel

p yo

uto

rem

embe

r th

e m

ean

ing

of a

un

ifor

mte

ssel

lati

on?

Sam

ple

an

swer

:Th

ree

sim

ilar

mea

nin

gs

of

un

iform

are

unv

aryi

ng

,co

nsi

sten

t,an

d id

enti

cal.

In a

un

iform

tes

sella

tio

n,t

he

arra

ng

emen

t o

f sh

apes

an

d a

ng

les

at e

ach

vert

ex is

th

e sa

me.

Eac

h o

f th

e th

ree

ever

yday

mea

nin

gs

des

crib

es t

his

situ

atio

n.

©G

lenc

oe/M

cGra

w-H

ill50

2G

lenc

oe G

eom

etry

Po

lyg

on

al N

um

ber

sC

erta

in n

um

bers

rel

ated

to

regu

lar

poly

gon

s ar

e ca

lled

pol

ygon

al n

um

ber

s.T

he

char

tsh

ows

seve

ral

tria

ngu

lar,

squ

are,

and

pen

tago

nal

nu

mbe

rs.T

he

ran

kof

a p

olyg

on n

um

ber

is t

he

nu

mbe

r of

dot

s on

eac

h “

side

”of

th

e ou

ter

poly

gon

.For

exa

mpl

e,th

e pe

nta

gon

aln

um

ber

22 h

as a

ran

k of

4.

Pol

ygon

al n

um

bers

can

be

desc

ribe

d w

ith

for

mu

las.

For

exa

mpl

e,a

tria

ngu

lar

nu

mbe

r T

of r

ank

r ca

n b

e de

scri

bed

by T

��r(

r2�

1)�

.

An

swer

eac

h q

ues

tion

.

1.D

raw

a d

iagr

am t

o fi

nd

the

2.D

raw

a d

iagr

am t

o fi

nd

the

tria

ngu

lar

nu

mbe

r of

ran

k 5.

15pe

nta

gon

al n

um

ber

of r

ank

5.35

3.W

rite

a f

orm

ula

for

a s

quar

e n

um

ber

4.W

rite

a f

orm

ula

for

a p

enta

gon

alS

of r

ank

r.n

um

ber

Pof

ran

k r.

S�

r2P

��r(

3r2�

1)�

5.W

hat

is

the

ran

k of

th

e pe

nta

gon

al6.

Lis

t th

e h

exag

onal

nu

mbe

rs f

or r

anks

nu

mbe

r 70

?7

1 to

5.(

Hin

t:D

raw

a d

iagr

am.)

1,6,

15,2

8,45

Ran

k 1

Tria

ngle

13

610

14

916

15

1222

Squ

are

Pen

tago

n

Ran

k 2

Ran

k 3

Ran

k 4

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-4

9-4



Stu

dy G

uid

e a

nd I

nte

rven

tion

Dila

tio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-5

9-5

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etry

Lesson 9-5

Cla

ssif

y D

ilati

on

sA

dil

atio

nis

a t

ran

sfor

mat

ion

in

wh

ich

th

e im

age

may

be

adi

ffer

ent

size

th

an t

he

prei

mag

e.A

dil

atio

n r

equ

ires

a c

ente

r po

int

and

a sc

ale

fact

or,r

.

Let

rre

pres

ent

the

scal

e fa

ctor

of

a di

latio

n.

If | r

| �1,

the

n th

e di

latio

n is

an

enla

rgem

ent.

If | r

| �1,

the

n th

e di

latio

n is

a c

ongr

uenc

e tr

ansf

orm

atio

n.

If 0

| r

| 1,

the

n th

e di

latio

n is

a r

educ

tion.

Dra

w t

he

dil

atio

n i

mag

e of

�

AB

Cw

ith

cen

ter

Oan

d r

�2.

Dra

w O

A� �

� ,O

B��

� ,an

d O

C��

� .L

abel

poi

nts

A�,

B�,

and

C�

so t

hat

OA�

�2(

OA

),O

B�

�2(

OB

),an

d O

C�

�2(

OC

).�

A�B

�C�

is a

dil

atio

n o

f �

AB

C.

Dra

w t

he

dil

atio

n i

mag

e of

eac

h f

igu

re w

ith

cen

ter

Can

d t

he

give

n s

cale

fac

tor.

Des

crib

e ea

ch t

ran

sfor

mat

ion

as

an e

nla

rgem

ent,

con

gru

ence

,or

red

uct

ion

.

1.r

�2

enla

rgem

ent

2.r

��1 2�

red

uct

ion

3.r

�1

con

gru

ence

4.r

�3

enla

rgem

ent

5.r

��2 3�

red

uct

ion

6.r

�1

con

gru

ence

KJ

HF

C

G

M C

P

R V

W

M�

P� R

� V�

W�

CS

TS

�

T�

C

W Y

XZ

C

ST

RS

�T

�

R�

CB

A A�

B�

O

B A

CO

B A

C

A�

B�

C�

Exer

cises

Exer

cises

Exam

ple

Exam

ple

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etry

Iden

tify

th

e Sc

ale

Fact

or

If y

ou k

now

cor

resp

ondi

ng

mea

sure

men

ts f

or a

pre

imag

ean

d it

s di

lati

on i

mag

e,yo

u c

an f

ind

the

scal

e fa

ctor

.

Det

erm

ine

the

scal

e fa

ctor

for

th

e d

ilat

ion

of

X �Y�

to A�

B�.D

eter

min

e w

het

her

th

e d

ilat

ion

is

an e

nla

rgem

ent,

red

uct

ion

,or

con

gru

ence

tra

nsf

orm

ati

on.

scal

e fa

ctor

�� pr

i em imagae gele

ln eg nt gh th�

��8 4

u un n

i it ts s�

�2

Th

e sc

ale

fact

or i

s gr

eate

r th

an 1

,so

the

dila

tion

is

an e

nla

rgem

ent.

Det

erm

ine

the

scal

e fa

ctor

for

eac

h d

ilat

ion

wit

h c

ente

r C

.Det

erm

ine

wh

eth

er t

he

dil

atio

n i

s an

en

larg

emen

t,re

du

ctio

n,o

r co

ng

ruen

ce t

ran

sfor

ma

tion

.

1.C

GH

Jis

a d

ilat

ion

im

age

of C

DE

F.

2.�

CK

Lis

a d

ilat

ion

im

age

of �

CK

L.

2;en

larg

emen

t1;

con

gru

ence

3.S

TU

VW

Xis

a d

ilat

ion

im

age

4.�

CP

Qis

a d

ilat

ion

im

age

of �

CY

Z.

of M

NO

PQ

R.

�1 2� ;re

du

ctio

n�1 3� ;

red

uct

ion

5.�

EF

Gis

a d

ilat

ion

im

age

of �

AB

C.

6.�

HJ

Kis

a d

ilat

ion

im

age

of �

HJ

K.

1.5;

enla

rgem

ent

1;co

ng

ruen

ce

C

J

K

H

CB

DG

F

EA

C

Z

Y

P

QC

MN

O

PQ

RU

X

ST V

W

CKL

G CD

H

FE

J

A B

X Y

C

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Dila

tio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-5

9-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Dila

tio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-5

9-5

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etry

Lesson 9-5

Dra

w t

he

dil

atio

n i

mag

e of

eac

h f

igu

re w

ith

cen

ter

Can

d t

he

give

n s

cale

fac

tor.

1.r

�2

2.r

��1 4�

Fin

d t

he

mea

sure

of

the

dil

atio

n i

mag

e M�

��N��o

r of

th

e p

reim

age

M�N�

usi

ng

the

give

n s

cale

fac

tor.

3.M

N�

3,r

�3

4.M

�N�

�7,

r�

21

M�N

��

9M

N�

�1 3�

CO

OR

DIN

ATE

GEO

MET

RYF

ind

th

e im

age

of e

ach

pol

ygon

,giv

en t

he

vert

ices

,aft

era

dil

atio

n c

ente

red

at

the

orig

in w

ith

a s

cale

fac

tor

of 2

.Th

en g

rap

h a

dil

atio

n

cen

tere

d a

t th

e or

igin

wit

h a

sca

le f

acto

r of

�1 2� .

5.J

(2,4

),K

(4,4

),P

(3,2

)6.

D(�

2,0)

,G(0

,2),

F(2

,�2)

Det

erm

ine

the

scal

e fa

ctor

for

eac

h d

ilat

ion

wit

h c

ente

r C

.Det

erm

ine

wh

eth

er t

he

dil

atio

n i

s an

en

larg

emen

t,re

du

ctio

n,o

r co

ng

ruen

ce t

ran

sfor

ma

tion

.Th

e d

ash

edfi

gure

is

the

dil

atio

n i

mag

e.

7.8.

4;en

larg

emen

t1;

con

gru

ence

tra

nsf

orm

atio

n

CC

G�

F�

D�

G�

F�

D�

x

y

O

J�K

�

P�

J�

K�

P�

x

y

O

CC

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etry

Dra

w t

he

dil

atio

n i

mag

e of

eac

h f

igu

re w

ith

cen

ter

Can

d t

he

give

n s

cale

fac

tor.

1.r

��3 2�

2.r

��2 3�

Fin

d t

he

mea

sure

of

the

dil

atio

n i

mag

e A�

��T��o

r of

th

e p

reim

age

A�T�

usi

ng

the

give

nsc

ale

fact

or.

3.A

T�

15,r

��3 5�

4.A

T�

30,r

��

�1 6�5.

A�T

��

12,r

��4 3�

A�T

��

9A

�T�

�5

AT

�9

CO

OR

DIN

ATE

GEO

MET

RYF

ind

th

e im

age

of e

ach

pol

ygon

,giv

en t

he

vert

ices

,aft

era

dil

atio

n c

ente

red

at

the

orig

in w

ith

a s

cale

fac

tor

of 2

.Th

en g

rap

h a

dil

atio

n

cen

tere

d a

t th

e or

igin

wit

h a

sca

le f

acto

r of

�1 2� .

6.A

(1,1

),C

(2,3

),D

(4,2

),E

(3,1

)7.

Q(�

1,�

1),R

(0,2

),S

(2,1

)

Det

erm

ine

the

scal

e fa

ctor

for

eac

h d

ilat

ion

wit

h c

ente

r C

.Det

erm

ine

wh

eth

er t

he

dil

atio

n i

s an

en

larg

emen

t,re

du

ctio

n,o

r co

ng

ruen

ce t

ran

sfor

ma

tion

.Th

e d

otte

dfi

gure

is

the

dil

atio

n i

mag

e.

8.�1 3� ;

red

uct

ion

9.2;

enla

rgem

ent

10.P

HO

TOG

RA

PHY

Est

ebe

enla

rged

a 4

-in

ch b

y 6-

inch

ph

otog

raph

by

a fa

ctor

of

�5 2� .W

hat

are

the

new

dim

ensi

ons

of t

he

phot

ogra

ph?

10 in

.by

15 in

.

CC

S�

Q�

R�

S�

Q�

R�

x

y

O

C�

D�

E�

A�

C�

D�

E�

A�

x

y

O

C

C

Pra

ctic

e (

Ave

rag

e)

Dila

tio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-5

9-5



Readin

g t

o L

earn

Math

em

ati

csD

ilati

on

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-5

9-5

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etry

Lesson 9-5

Pre-

Act

ivit

yH

ow d

o yo

u u

se d

ilat

ion

s w

hen

you

use

a c

omp

ute

r?

Rea

d th

e in

trod

uct

ion

to

Les

son

9-5

at

the

top

of p

age

490

in y

our

text

book

.

In a

ddit

ion

to

the

exam

ple

give

n i

n y

our

text

book

,giv

e tw

o ev

eryd

ayex

ampl

es o

f sc

alin

g an

obj

ect,

one

that

mak

es t

he

obje

ct l

arge

r an

d an

oth

erth

at m

akes

it

smal

ler.

Sam

ple

an

swer

:la

rger

:en

larg

ing

ap

ho

tog

rap

h;

smal

ler:

mak

ing

a s

cale

mo

del

of

a bu

ildin

gR

ead

ing

th

e Le

sso

n1.

Eac

h o

f th

e va

lues

of

rgi

ven

bel

ow r

epre

sen

ts t

he

scal

e fa

ctor

for

a d

ilat

ion

.In

eac

hca

se,d

eter

min

e w

het

her

th

e di

lati

on i

s an

en

larg

emen

t,a

red

uct

ion

,or

a co

ngr

uen

cetr

ansf

orm

atio

n.

a.r

�3

enla

rgem

ent

b.r

�0.

5re

du

ctio

nc.

r�

�0.

75re

du

ctio

nd

.r�

�1

con

gru

ence

tra

nsf

orm

atio

ne.

r�

�2 3�re

du

ctio

nf.

r�

��3 2�

enla

rgem

ent

g.r

��

1.01

enla

rgem

ent

h.r

�0.

999

red

uct

ion

2.D

eter

min

e w

het

her

eac

h s

ente

nce

is

alw

ays,

som

etim

es,o

r n

ever

tru

e.If

th

e se

nte

nce

is

not

alw

ays

tru

e,ex

plai

n w

hy.

Fo

r ex

pla

nat

ion

s,sa

mp

le a

nsw

ers

are

giv

en.

a.A

dil

atio

n r

equ

ires

a c

ente

r po

int

and

a sc

ale

fact

or.

alw

ays

b.

A d

ilat

ion

ch

ange

s th

e si

ze o

f a

figu

re.

So

met

imes

;if

th

e d

ilati

on

is a

con

gru

ence

tra

nsf

orm

atio

n,t

he

size

of

the

fig

ure

is u

nch

ang

ed.

c.A

dil

atio

n c

han

ges

the

shap

e of

a f

igu

re.

Nev

er;

all d

ilati

on

s p

rod

uce

sim

ilar

fig

ure

s,an

d s

imila

r fi

gu

res

hav

e th

e sa

me

shap

e.d

.T

he

imag

e of

a f

igu

re u

nde

r a

dila

tion

lie

s on

th

e op

posi

te s

ide

of t

he

cen

ter

from

th

epr

eim

age.

So

met

imes

;th

is is

on

ly t

rue

wh

en t

he

scal

e fa

cto

r is

neg

ativ

e.e.

A s

imil

arit

y tr

ansf

orm

atio

n i

s a

con

gru

ence

tra

nsf

orm

atio

n.

So

met

imes

;th

is is

tru

e o

nly

wh

en t

he

scal

e fa

cto

r is

1 o

r �

1.f.

Th

e ce

nte

r of

a d

ilat

ion

is

its

own

im

age.

alw

ays

g.A

dil

atio

n i

s an

iso

met

ry.

So

met

imes

;th

is is

tru

e o

nly

wh

en t

he

dila

tio

n is

a co

ng

ruen

ce t

ran

sfo

rmat

ion

.h

.T

he

scal

e fa

ctor

for

a d

ilat

ion

is

a po

siti

ve n

um

ber.

So

met

imes

;th

e sc

ale

fact

or

can

be

any

po

siti

ve o

r n

egat

ive

nu

mb

er.

i.D

ilat

ion

s pr

odu

ce s

imil

ar f

igu

res.

alw

ays

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r so

met

hin

g is

to

expl

ain

it

to s

omeo

ne

else

.Su

ppos

e th

at y

our

clas

smat

e L

ydia

is

hav

ing

trou

ble

un

ders

tan

din

g th

e re

lati

onsh

ip b

etw

een

sim

ilar

ity

tran

sfor

mat

ion

san

d co

ngr

uen

ce t

ran

sfor

mat

ion

s.H

ow c

an y

ou e

xpla

in t

his

to

her

?S

amp

le a

nsw

er:

A c

on

gru

ence

tra

nsf

orm

atio

n p

rese

rves

th

e si

ze a

nd

shap

e o

f a

fig

ure

,th

at is

,th

e im

age

is c

on

gru

ent

to t

he

pre

imag

e.A

sim

ilari

ty t

ran

sfo

rmat

ion

pre

serv

es t

he

shap

e o

f a

fig

ure

,th

at is

,th

eim

age

is s

imila

r to

th

e p

reim

age.

A s

imila

rity

tra

nsf

orm

atio

n is

aco

ng

ruen

ce t

ran

sfo

rmat

ion

if t

he

scal

e fa

cto

r is

1 o

r �

1.

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Sim

ilar

Cir

cles

You

may

be

surp

rise

d to

lea

rn t

hat

tw

o n

onco

ngr

uen

t ci

rcle

s th

at l

ie i

nth

e sa

me

plan

e an

d h

ave

no

com

mon

in

teri

or p

oin

ts c

an b

e m

appe

don

e on

to t

he

oth

er b

y m

ore

than

on

e di

lati

on.

1.H

ere

is d

iagr

am t

hat

su

gges

ts o

ne

way

to

m

ap a

sm

alle

r ci

rcle

on

to a

lar

ger

one

usi

ng

a di

lati

on.T

he

circ

les

are

give

n.T

he

lin

essu

gges

t h

ow t

o fi

nd

the

cen

ter

for

the

dila

tion

.Des

crib

e h

ow t

he

cen

ter

is f

oun

d.U

se s

egm

ents

in

th

e di

agra

m t

o n

ame

the

scal

e fa

ctor

.

Dra

w t

he

two

co

mm

on

ext

ern

alta

ng

ents

.Th

e p

oin

t w

her

e th

ey in

ters

ect

is t

he

cen

ter

of

a d

ilati

on

th

at m

aps

�A

on

to �

A�;

scal

e fa

cto

r �

�A A�M M�

�.

2.H

ere

is a

not

her

pai

r of

non

con

gru

ent

circ

les

wit

h n

o co

mm

onin

teri

or p

oin

t.F

rom

Exe

rcis

e 1,

you

kn

ow y

ou c

an l

ocat

e a

poin

t of

f to

th

e le

ft o

f th

e sm

alle

r ci

rcle

th

at i

s th

e ce

nte

r fo

r a

dila

tion

map

pin

g �

Con

to �

C�.

Fin

d an

oth

er c

ente

r fo

r an

oth

er d

ilat

ion

that

map

s �

Con

to �

C�.

Mar

k an

d la

bel

segm

ents

to

nam

e th

esc

ale

fact

or.

Fin

d t

he

inte

rsec

tio

n o

f th

e tw

o c

om

mo

n in

tern

al t

ang

ents

;sc

ale

fact

or:

��C C�X X

��

.

X

X�

C�

CO

A �

M�

M

AO

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-5

9-5



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Vec

tors

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-6

9-6

©G

lenc

oe/M

cGra

w-H

ill50

9G

lenc

oe G

eom

etry

Lesson 9-6

Mag

nit

ud

e an

d D

irec

tio

nA

vec

tor

is a

dir

ecte

d se

gmen

t re

pres

enti

ng

a qu

anti

ty t

hat

has

bot

h m

agn

itu

de,

or l

engt

h,

and

dir

ecti

on.F

or e

xam

ple,

the

spee

d an

d di

rect

ion

of

an

airp

lan

e ca

n b

e re

pres

ente

d by

a v

ecto

r.In

sym

bols

,a v

ecto

r is

w

ritt

en a

s A

B�

,wh

ere

Ais

th

e in

itia

l po

int

and

Bis

th

e en

dpoi

nt,

or a

s v �

.

A v

ecto

r in

sta

nd

ard

pos

itio

nh

as i

ts i

nit

ial

poin

t at

(0,

0) a

nd

can

be

repr

esen

ted

by t

he

orde

red

pair

for

poi

nt

B.T

he

vect

or

at t

he

righ

t ca

n b

e ex

pres

sed

as v �

��5

,3�.

You

can

use

th

e D

ista

nce

For

mu

la t

o fi

nd

the

mag

nit

ude

| A

B�

| of

a ve

ctor

.You

can

des

crib

e th

e di

rect

ion

of

a ve

ctor

by

mea

suri

ng

the

angl

e th

at t

he

vect

or f

orm

s w

ith

th

e po

siti

vex-

axis

or

wit

h a

ny

oth

er h

oriz

onta

l li

ne.

Fin

d t

he

mag

nit

ud

e an

d d

irec

tion

of

AB�

for

A(5

,2)

and

B(8

,7).

Fin

d th

e m

agn

itu

de.

| AB

| ��

(x2

��

x 1)2

��

(y2

��

y 1)2

��

�(8

�5

�)2

�(

�7

�2

�)2 �

��

34�or

abo

ut

5.8

un

its

To

fin

d th

e di

rect

ion

,use

th

e ta

nge

nt

rati

o.

tan

A�

�5 3�T

he t

ange

nt r

atio

is o

ppos

ite o

ver

adja

cent

.

m�

A�

59.0

Use

a c

alcu

lato

r.

Th

e m

agn

itu

de o

f th

e ve

ctor

is

abou

t 5.

8 u

nit

s an

d it

s di

rect

ion

is

59°.

Fin

d t

he

mag

nit

ud

e an

d d

irec

tion

of

AB�

for

the

give

n c

oord

inat

es.R

oun

d t

o th

en

eare

st t

enth

.

1.A

(3,1

),B

(�2,

3)2.

A(0

,0),

B(�

2,1)

5.4;

158.

2°2.

2;15

3.4°

3.A

(0,1

),B

(3,5

)4.

A(�

2,2)

,B(3

,1)

5;53

.1°

5.1;

348.

7°

5.A

(3,4

),B

(0,0

)6.

A(4

,2),

B(0

,3)

5;23

3.1°

4.1;

166.

0°

x

y

O

B( 8

,7)

A( 5

, 2)

x

y

O

B( 5

, 3)

A( 0

, 0)

V�

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill51

0G

lenc

oe G

eom

etry

Tran

slat

ion

s w

ith

Vec

tors

Rec

all

that

th

e tr

ansf

orm

atio

n (

a,b)

→(a

�2,

b�

3)re

pres

ents

a t

ran

slat

ion

rig

ht

2 u

nit

s an

d do

wn

3 u

nit

s.T

he

vect

or �

2,�

3�is

an

oth

er w

ay t

ode

scri

be t

hat

tra

nsl

atio

n.A

lso,

two

vect

ors

can

be

adde

d:�a

,b�

��c

,d�

��a

�c,

b�

d�.

Th

esu

m o

f tw

o ve

ctor

s is

cal

led

the

resu

ltan

t.

Gra

ph

th

e im

age

of p

aral

lelo

gram

RS

TU

un

der

th

e tr

ansl

atio

n b

y th

e ve

ctor

s m �

��3

,�1�

and

n �

��

2,�

4�.

Fin

d th

e su

m o

f th

e ve

ctor

s.m�

�n�

��3

,�1�

��

2,�

4��

�3 �

2,�

1 �

4��

�1,�

5�T

ran

slat

e ea

ch v

erte

x of

par

alle

logr

am R

ST

Uri

ght

1 u

nit

an

d do

wn

5 u

nit

s.

Gra

ph

th

e im

age

of e

ach

fig

ure

un

der

a t

ran

slat

ion

by

the

give

n v

ecto

r(s)

.

1.�

AB

Cw

ith

ver

tice

s A

(�1,

2),B

(0,0

),2.

AB

CD

wit

h v

erti

ces

A(�

4,1)

,B(�

2,3)

,an

d C

(2,3

);m �

��2

,�3�

C(1

,1),

and

D(�

1,�

1);n�

��3

�3�

3.A

BC

Dw

ith

ver

tice

s A

(�3,

3),B

(1,3

),C

(1,1

),an

d D

(�3,

1);t

he

sum

of

p ��

��2,

1�an

dq�

��5

,�4�

Giv

en m �

��1

,�2�

and

n��

��3,

�4�

,rep

rese

nt

each

of

the

foll

owin

g as

a s

ingl

eve

ctor

.

4.m �

�n�

5.n�

�m�

��2,

�6�

��4,

�2�

x

y

D�

A�

B� C

�

DAB C

x

y

D�

A�

B�

C�

D

A

B

C

x

y A�

B�C

�

A

B

C

x

y

R�S

�

T�

U�

R

S

TU

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Vec

tors

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-6

9-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Vec

tors

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-6

9-6

©G

lenc

oe/M

cGra

w-H

ill51

1G

lenc

oe G

eom

etry

Lesson 9-6

Wri

te t

he

com

pon

ent

form

of

each

vec

tor.

1.2.

�3,4

��5

,�5�

Fin

d t

he

mag

nit

ud

e an

d d

irec

tion

of

RS�

for

the

give

n c

oord

inat

es.R

oun

d t

o th

en

eare

st t

enth

.

3.R

(2,�

3),S

(4,9

)4.

R(0

,2),

S(3

,12)

2�37�

�12

.2,8

0.5°

�10

9�

�10

.4,7

3.3°

5.R

(5,4

),S

(�3,

1)6.

R(1

,5),

S(�

4,�

6)

�73�

�8.

5,20

0.6°

�14

6�

�12

.1,2

45.6

°

Gra

ph

th

e im

age

of e

ach

fig

ure

un

der

a t

ran

slat

ion

by

the

give

n v

ecto

r(s)

.

7.�

AB

Cw

ith

ver

tice

s A

(�4,

3),B

(�1,

4),

8.tr

apez

oid

wit

h v

erti

ces

T(�

4,�

2),

C(�

1,1)

;t��

�4,�

3�R

(�1,

�2)

,S(�

2,�

3),Y

(�3,

�3)

;a��

�3,1

�an

d b�

��2

,4�

Fin

d t

he

mag

nit

ud

e an

d d

irec

tion

of

each

res

ult

ant

for

the

give

n v

ecto

rs.

9.y �

��7

,0�,

z��

�0,6

�10

.b��

�3,2

�,c�

��

2,3�

�85�

�9.

2,40

.6°

�26�

�5.

1,78

.7°

S�

Y�

T�

R�

SY

TR

x

y

O

C�

B�

A�

CB

A

x

y

O

x

y

O

B( –

2, 2

)

D( 3

, –3)

x

y

OC

( 1, –

1)

E( 4

, 3)

©G

lenc

oe/M

cGra

w-H

ill51

2G

lenc

oe G

eom

etry

Wri

te t

he

com

pon

ent

form

of

each

vec

tor.

1.2.

�6,4

��5

,�8�

Fin

d t

he

mag

nit

ud

e an

d d

irec

tion

of

FG�

for

the

give

n c

oord

inat

es.R

oun

d t

o th

en

eare

st t

enth

.

3.F

(�8,

�5)

,G(�

2,7)

4.F

(�4,

1),G

(5,�

6)

6�5�

�13

.4,6

3.4°

�13

0�

�11

.4,3

22.1

°

Gra

ph

th

e im

age

of e

ach

fig

ure

un

der

a t

ran

slat

ion

by

the

give

n v

ecto

r(s)

.

5.�

QR

Tw

ith

ver

tice

s Q

(�1,

1),R

(1,4

),6.

trap

ezoi

d w

ith

ver

tice

s J

(�4,

�1)

,T

(5,1

);s �

��

2,�

5�K

(0,�

1),L

(�1,

�3)

,M(�

2,�

3);c�

��5

,4�

and

d��

��2,

1�

Fin

d t

he

mag

nit

ud

e an

d d

irec

tion

of

each

res

ult

ant

for

the

give

n v

ecto

rs.

7.a�

��

6,4�

,b��

�4,6

�8.

e��

��4,

�5�

,f��

��1,

3�

2�26�

�10

.2,1

01.3

°�

29��

5.4,

201.

8°

AV

IATI

ON

For

Exe

rcis

es 9

an

d 1

0,u

se t

he

foll

owin

g in

form

atio

n.

A je

t be

gins

a f

light

alo

ng a

pat

h du

e no

rth

at 3

00 m

iles

per

hour

.A w

ind

is b

low

ing

due

wes

tat

30

mile

s pe

r ho

ur.

9.F

ind

the

resu

ltan

t ve

loci

ty o

f th

e pl

ane.

abo

ut

301.

5 m

ph

10.F

ind

the

resu

ltan

t di

rect

ion

of

the

plan

e.ab

ou

t 5.

7°w

est

of

du

e n

ort

h

L�M

�

J�K

�

LM

JK

x

y

O

T�

R�

Q�

T

R

Qx

y

O

x

y

O

K( –

2, 4

)

L(3,

–4)

x

y

O

A( –

2, –

2)

B( 4

, 2)

Pra

ctic

e (

Ave

rag

e)

Vec

tors

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-6

9-6



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csV

ecto

rs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-6

9-6

©G

lenc

oe/M

cGra

w-H

ill51

3G

lenc

oe G

eom

etry

Lesson 9-6

Pre-

Act

ivit

yH

ow d

o ve

ctor

s h

elp

a p

ilot

pla

n a

fli

ght?

Rea

d th

e in

trod

uct

ion

to

Les

son

9-6

at

the

top

of p

age

498

in y

our

text

book

.

Wh

y do

pil

ots

ofte

n h

ead

thei

r pl

anes

in

a s

ligh

tly

diff

eren

t di

rect

ion

fro

mth

eir

dest

inat

ion

?S

amp

le a

nsw

er:

to c

om

pen

sate

fo

r th

e ef

fect

of

the

win

d s

o t

hat

th

e re

sult

will

be

that

th

e p

lan

e w

ill a

ctu

ally

fly

in t

he

corr

ect

dir

ecti

on

Rea

din

g t

he

Less

on

1.S

upp

ly t

he

mis

sin

g w

ords

or

phra

ses

to c

ompl

ete

the

foll

owin

g se

nte

nce

s.

a.A

is

a d

irec

ted

segm

ent

repr

esen

tin

g a

quan

tity

th

at h

as b

oth

mag

nit

ude

an

d di

rect

ion

.

b.

Th

e le

ngt

h o

f a

vect

or i

s ca

lled

its

.

c.T

wo

vect

ors

are

para

llel

if

and

only

if

they

hav

e th

e sa

me

or

dire

ctio

n.

d.

A v

ecto

r is

in

if

it

is d

raw

n w

ith

in

itia

l po

int

at t

he

orig

in.

e.T

wo

vect

ors

are

equ

al i

f an

d on

ly i

f th

ey h

ave

the

sam

e an

d th

e

sam

e.

f.T

he

sum

of

two

vect

ors

is c

alle

d th

e .

g.A

vec

tor

is w

ritt

en i

n

if i

t is

exp

ress

ed a

s an

ord

ered

pai

r.

h.

Th

e pr

oces

s of

mu

ltip

lyin

g a

vect

or b

y a

con

stan

t is

cal

led

.

2.W

rite

eac

h v

ecto

r de

scri

bed

belo

w i

n c

ompo

nen

t fo

rm.

a.a

vect

or i

n s

tan

dard

pos

itio

n w

ith

en

dpoi

nt

(a,b

)�a

,b�

b.

a ve

ctor

wit

h i

nit

ial

poin

t (a

,b)

and

endp

oin

t (c

,d)

�c�

a,d

�b

�c.

a ve

ctor

in

sta

nda

rd p

osit

ion

wit

h e

ndp

oin

t (�

3,5)

��3,

5�d

.a

vect

or w

ith

in

itia

l po

int

(2,�

3) a

nd

endp

oin

t (6

,�8)

�4,�

5�e.

a��

b�if

a��

��3,

5�an

d b�

��6

,�4�

�3,1

�f.

5u�if

u��

�8,�

6��4

0,�

30�

g.�

�1 3� v�if

v��

��15

,24�

�5,�

8�h

.0.

5u��

1.5v�

if u�

��1

0,�

10�

and

v��

��8,

6��

7,4�

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al t

erm

is

to r

elat

e it

to

a te

rm y

ou a

lrea

dykn

ow.Y

ou le

arne

d ab

out

scal

e fa

ctor

sw

hen

you

stud

ied

sim

ilari

ty a

nd d

ilati

ons.

How

is t

heid

ea o

f a

scal

arre

late

d to

sca

le f

acto

rs?

Sam

ple

an

swer

:A s

cala

ris

th

e te

rm u

sed

for

a co

nst

ant

(a s

pec

ific

real

nu

mb

er)

wh

en w

ork

ing

with

vec

tors

.A v

ecto

rh

as b

oth

mag

nit

ud

e an

d d

irec

tio

n,w

hile

a s

cala

r is

just

a m

agn

itu

de.

Mu

ltip

lyin

g a

vec

tor

by a

po

siti

ve s

cala

r ch

ang

es t

he

mag

nit

ud

e o

f th

eve

cto

r,bu

t n

ot

the

dir

ecti

on

,so

it r

epre

sen

ts a

ch

ang

e in

sca

le.

scal

ar m

ult

iplic

atio

nco

mp

on

ent

formre

sult

ant

dir

ecti

on

mag

nit

ud

est

and

ard

po

siti

on

op

po

site

mag

nit

ud

e

vect

or

©G

lenc

oe/M

cGra

w-H

ill51

4G

lenc

oe G

eom

etry

Rea

din

g M

ath

emat

ics

Man

y qu

anti

ties

in

nat

ure

can

be

thou

ght

of a

s ve

ctor

s.T

he

scie

nce

of

phys

ics

invo

lves

man

y ve

ctor

qu

anti

ties

.In

rea

din

g ab

out

appl

icat

ion

sof

mat

hem

atic

s,as

k yo

urs

elf

wh

eth

er t

he

quan

titi

es i

nvo

lve

only

mag

nit

ude

or

both

mag

nit

ude

an

d di

rect

ion

.Th

e fi

rst

kin

d of

qu

anti

tyis

cal

led

scal

ar.T

he

seco

nd

kin

d is

a v

ecto

r.

Cla

ssif

y ea

ch o

f th

e fo

llow

ing.

Wri

te s

cala

ror

vec

tor.

1.th

e m

ass

of a

boo

ksc

alar

2.a

car

trav

elin

g n

orth

at

55 m

phve

cto

r

3.a

ball

oon

ris

ing

24 f

eet

per

min

ute

vect

or

4.th

e si

ze o

f a

shoe

scal

ar

5.a

room

tem

pera

ture

of

22 d

egre

es C

elsi

us

scal

ar

6.a

wes

t w

ind

of 1

5 m

phve

cto

r

7.th

e ba

ttin

g av

erag

e of

a b

aseb

all

play

ersc

alar

8.a

car

trav

elin

g 60

mph

vect

or

9.a

rock

fal

lin

g at

10

mph

vect

or

10.y

our

age

scal

ar

11.t

he

forc

e of

Ear

th’s

gra

vity

act

ing

on a

mov

ing

sate

llit

eve

cto

r

12.t

he

area

of

a re

cord

rot

atin

g on

a t

urn

tabl

esc

alar

13.t

he

len

gth

of

a ve

ctor

in

th

e co

ordi

nat

e pl

ane

scal

ar

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-6

9-6



Stu

dy G

uid

e a

nd I

nte

rven

tion

Tran

sfo

rmat

ion

s w

ith

Mat

rice

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-7

9-7

©G

lenc

oe/M

cGra

w-H

ill51

5G

lenc

oe G

eom

etry

Lesson 9-7

Tran

slat

ion

s an

d D

ilati

on

sA

vec

tor

can

be

repr

esen

ted

by t

he

orde

red

pair

�x,

y�or

by t

he

colu

mn

mat

rix

.Wh

en t

he

orde

red

pair

s fo

r al

l th

e ve

rtic

es o

f a

poly

gon

are

plac

ed t

oget

her

,th

e re

sult

ing

mat

rix

is c

alle

d th

e ve

rtex

mat

rix

for

the

poly

gon

.

For

�A

BC

wit

h A

(�2,

2),B

(2,1

),an

d C

(�1,

�1)

,th

e ve

rtex

mat

rix

for

the

tria

ngl

e is

.

For

�A

BC

abov

e,u

se a

mat

rix

to f

ind

th

e co

ord

inat

es o

f th

eve

rtic

es o

f th

e im

age

of �

AB

Cu

nd

er t

he

tran

slat

ion

(x,

y) →

(x�

3,y

�1)

.T

o tr

ansl

ate

the

figu

re 3

un

its

to t

he

righ

t,ad

d 3

to e

ach

x-c

oord

inat

e.T

o tr

ansl

ate

the

figu

re 1

un

it d

own

,add

�1

to e

ach

y-c

oord

inat

e.V

erte

x M

atri

xT

ran

slat

ion

V

erte

x M

atri

xof

�A

BC

Mat

rix

of �

A�B

�C�

��

Th

e co

ordi

nat

es a

re A

�(1,

1),B

�(5,

0),a

nd

C�(

2,�

2).

For

�A

BC

abov

e,u

se a

mat

rix

to f

ind

th

e co

ord

inat

es o

f th

e ve

rtic

esof

th

e im

age

of �

AB

Cfo

r a

dil

atio

n c

ente

red

at

the

orig

in w

ith

sca

le f

acto

r 3.

Sca

leV

erte

x M

atri

xV

erte

x M

atri

xFa

ctor

of �

AB

Cof

�A

�B�C

�

3

�

Th

e co

ordi

nat

es a

re A

�(�

6,6)

,B�(

6,3)

,an

d C

�(�

3,�

3).

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n t

ran

slat

ion

s or

dil

atio

ns.

1.�

AB

Cw

ith

A(3

,1),

B(�

2,4)

,C(�

2,�

1);(

x,y)

→(x

�1,

y�

2)A

�(2,

3),B

�(�

3,6)

,C�(

�3,

1)2.

para

llel

ogra

m R

ST

Uw

ith

R(�

4,�

2),S

(�3,

1),T

(3,4

),U

(2,1

);(x

,y)

→(x

�4,

y�

3)

R�(

�8,

�5)

,S�(

�7,

�2)

,T�(

�1,

1),U

�(�

2,�

2)3.

rect

angl

e P

QR

Sw

ith

P(4

,0),

Q(3

,�3)

,R(�

3,�

1),S

(�2,

2);(

x,y)

→(x

�2,

y�

1)P

�(2,

1),Q

�(1,

�2)

,R�(

�5,

0),S

�(�

4,3)

4.�

AB

Cw

ith

A(�

2,�

1),B

(�2,

�3)

,C(2

,�1)

;dil

atio

n c

ente

red

at t

he

orig

in w

ith

sca

lefa

ctor

2A

�(�

4,�

2),B

�(�

4,�

6),C

�(4,

�2)

5.pa

rall

elog

ram

RS

TU

wit

h R

(4,�

2),S

(�4,

�1)

,T(�

2,3)

,U(6

,2);

dila

tion

cen

tere

d at

th

eor

igin

wit

h s

cale

fac

tor

1.5

R�(

6,�

3),S

�(�

6,�

1.5)

,T�(

�3,

4.5)

,U�(

9,3)

�6

6�

3

63

�3

�2

2�

1

21

�1

15

2 1

0�

2

33

3 �

1�

1�

1�

22

�1

2

1�

1

�2

2�

1

21

�1

x

y

B( 2

, 1)

C( –

1, –

1)

A( –

2, 2

)

x

y

Exer

cises

Exer

cises

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

©G

lenc

oe/M

cGra

w-H

ill51

6G

lenc

oe G

eom

etry

Ref

lect

ion

s an

d R

ota

tio

ns

Wh

en y

ou r

efle

ct a

n i

mag

e,on

e w

ay t

o fi

nd

the

coor

din

ates

of

the

refl

ecte

d ve

rtic

es i

s to

mu

ltip

ly t

he

vert

ex m

atri

x of

th

e ob

ject

by

are

flec

tion

mat

rix.

To

perf

orm

mor

e th

an o

ne

refl

ecti

on,m

ult

iply

by

one

refl

ecti

on m

atri

xto

fin

d th

e fi

rst

imag

e.T

hen

mu

ltip

ly b

y th

e se

con

d m

atri

x to

fin

d th

e fi

nal

im

age.

Th

em

atri

ces

for

refl

ecti

ons

in t

he

axes

,th

e or

igin

,an

d th

e li

ne

y�

xar

e sh

own

bel

ow.

Fo

r a

refl

ecti

on

in t

he:

x-ax

isy-

axis

ori

gin

line

y�

x

Mu

ltip

ly t

he

vert

ex

mat

rix

by:

�A

BC

has

ver

tice

s A

(�2,

3),B

(1,4

),an

d

C(3

,0).

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

th

e im

age

of �

AB

Caf

ter

a re

flec

tion

in

th

e x-

axis

.T

o re

flec

t in

th

e x-

axis

,mu

ltip

ly t

he

vert

ex m

atri

x of

�A

BC

by t

he

refl

ecti

on m

atri

x fo

r th

e x-

axis

.

Ref

lect

ion

Mat

rix

Ver

tex

Mat

rix

Ver

tex

Mat

rix

for

x-ax

isof

�A

BC

of �

A’B

’C’

�

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n r

efle

ctio

n.

1.�

AB

Cw

ith

A(�

3,2)

,B(�

1,3)

,C(1

,0);

refl

ecti

on i

n t

he

x-ax

is

A�(

�3,

�2)

,B�(

�1,

�3)

,C�(

1,0)

2.�

XY

Zw

ith

X(2

,�1)

,Y(4

,�3)

,Z(�

2,1)

;ref

lect

ion

in

th

e y-

axis

X�(

�2,

�1)

,Y�(

�4,

�3)

,Z�(

2,1)

3.�

AB

Cw

ith

A(3

,4),

B(�

1,0)

,C(�

2,4)

;ref

lect

ion

in

th

e or

igin

A�(

�3,

�4)

,B�(

1,0)

,C�(

2,�

4)

4.pa

rall

elog

ram

RS

TU

wit

h R

(�3,

2),S

(3,2

),T

(5,�

1),U

(�1,

�1)

;ref

lect

ion

in t

he li

ne y

�x

R�(

2,�

3),S

�(2,

3),T

�(�

1,5)

,U�(

�1,

�1)

5.�

AB

Cw

ith

A(2

,3),

B(�

1,2)

,C(1

,�1)

;ref

lect

ion

in

th

e or

igin

,th

en r

efle

ctio

n i

n t

he

lin

ey

�x

A�(

�3,

�2)

,B�(

�2,

1),C

�(1,

�1)

6.pa

rall

elog

ram

RS

TU

wit

h R

(0,2

),S

(4,2

),T

(3,�

2),U

(�1,

�2)

;ref

lect

ion

in

th

e x-

axis

,th

en r

efle

ctio

n i

n t

he

y-ax

is

R�(

0,�

2),S

�(�

4,�

2),T

�(�

3,2)

,U�(

1,2)

�2

13

�3

�4

0�

21

3

34

01

0 0

�1

x

y

A�

B�

AB

C

01

10

�1

0

0�

1�

10

0

11

00

�1

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Tran

sfo

rmat

ion

s w

ith

Mat

rice

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-7

9-7

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Tran

sfo

rmat

ion

s w

ith

Mat

rice

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-7

9-7

©G

lenc

oe/M

cGra

w-H

ill51

7G

lenc

oe G

eom

etry

Lesson 9-7

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n t

ran

slat

ion

s.

1.�

ST

Uw

ith

S(6

,4),

T(9

,7),

and

U(1

4,2)

;(x,

y) →

(x�

4,y

�3)

S�(

2,7)

,T�(

5,10

),U

�(10

,5)

2.�

GH

Iw

ith

G(�

5,0)

,H(�

3,6)

,an

d I(

�2,

1);(

x,y)

→(x

�2,

y�

6)

G�(

�3,

6),H

�(�

1,12

),I�

(0,7

)

Use

sca

lar

mu

ltip

lica

tion

to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

each

fig

ure

for

a d

ilat

ion

cen

tere

d a

t th

e or

igin

wit

h t

he

give

n s

cale

fac

tor.

3.�

DE

Fw

ith

D(2

,1),

E(5

,4),

and

F(7

,2);

r�

4

D�(

8,4)

,E�(

20,1

6),F

�(28

,8)

4.qu

adri

late

ral

WX

YZ

wit

h W

(�9,

6),X

(�6,

3),Y

(3,1

2),a

nd

Z(�

6,15

);r

��1 3�

W�(

�3,

2),X

�(�

2,1)

,Y�(

1,4)

,Z�(

�2,

5)

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n r

efle

ctio

n.

5.�

MN

Ow

ith

M(�

5,1)

,N(�

2,3)

,an

d O

(2,0

);y-

axis

M�(

5,1)

,N�(

2,3)

,O�(

�2,

0)

6.qu

adri

late

ral

AB

CD

wit

h A

(3,1

),B

(6,�

2),C

(5,�

5),a

nd

D(1

,�6)

;x-a

xis

A�(

3,�

1),B

�(6,

2),C

�(5,

5),D

�(1,

6)

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n r

otat

ion

.

7.�

RS

Tw

ith

R(�

2,�

2),S

(�3,

3),a

nd

T(2

,2);

90°c

oun

terc

lock

wis

e

R�(

2,�

2),S

�(�

3,�

3),T

�(�

2,2)

8.�

LM

NP

wit

h L

(3,4

),M

(7,4

),N

(9,�

3),a

nd

P(5

,�3)

;180

°co

un

terc

lock

wis

e

L�(

�3,

�4)

,M�(

�7,

�4)

,N�(

�9,

3),P

�(�

5,3)

©G

lenc

oe/M

cGra

w-H

ill51

8G

lenc

oe G

eom

etry

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n t

ran

slat

ion

s.

1.�

KL

Mw

ith

K(�

7,�

3),L

(4,9

),an

d M

(9,�

6);(

x,y)

→(x

�7,

y�

2)

K�(

�14

,�1)

,L�(

�3,

11),

M�(

2,�

4)

2.�

AB

CD

wit

h A

(�4,

3),B

(�2,

8),C

(3,1

0),a

nd

D(1

,5);

(x,y

) →

(x�

3,y

�9)

A�(

�1,

�6)

,B�(

1,�

1),C

�(6,

1),D

�(4,

�4)

Use

sca

lar

mu

ltip

lica

tion

to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

each

fig

ure

for

a d

ilat

ion

cen

tere

d a

t th

e or

igin

wit

h t

he

give

n s

cale

fac

tor.

3.qu

adri

late

ral

HIJ

Kw

ith

H(�

2,3)

,I(2

,6),

J(8

,3),

and

K(3

,�4)

;r�

��1 3�

H� ��2 3� ,

�1 �,

I��

�2 3� ,�

2 �,J

� ��8 3� ,

�1 �,

K� ��

1,�4 3� �

4.pe

nta

gon

DE

FG

Hw

ith

D(�

8,�

4),E

(�8,

2),F

(2,6

),G

(8,0

),an

d H

(4,�

6);r

��5 4�

D�(

�10

,�5)

,E� ��

10,�

5 2� �,F

� ��5 2� ,�1 25 �

�,G�(

10,0

),H

� �5,�

�1 25 ��

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n r

efle

ctio

n.

5.�

QR

Sw

ith

Q(�

5,�

4),R

(�1,

�1)

,an

d S

(2,�

6);x

-axi

s

Q�(

�5,

4),R

�(�

1,1)

,S�(

2,6)

6.qu

adri

late

ral

VX

YZ

wit

h V

(�4,

�2)

,X(�

3,4)

,Y(2

,1),

and

Z(4

,�3)

;y�

x

V�(

�2,

�4)

,X�(

4,�

3),Y

�(1,

2),Z

�(�

3,4)

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n r

otat

ion

.

7.�

EF

GH

wit

h E

(�5

�4)

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3,�

1),G

(5,�

1),a

nd

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,�4)

;90°

cou

nte

rclo

ckw

ise

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4,�

5),F

�(1,

�3)

,G�(

1,5)

,H�(

4,3)

8.qu

adri

late

ral

PS

TU

wit

h P

(�3,

5),S

(2,6

),T

(8,1

),an

d U

(�6,

�4)

;270

°co

un

terc

lock

wis

e

P�(

5,3)

,S�(

6,�

2),T

�(1,

�8)

,U�(

�4,

6)

9.FO

RES

TRY

A r

esea

rch

bot

anis

t m

appe

d a

sect

ion

of

fore

sted

lan

d on

a c

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e gr

idto

kee

p tr

ack

of e

nda

nge

red

plan

ts i

n t

he

regi

on.T

he

vert

ices

of

the

map

are

A(�

2,6)

,B

(9,8

),C

(14,

4),a

nd

D(1

,�1)

.Aft

er a

mon

th,t

he

bota

nis

t h

as d

ecid

ed t

o de

crea

se t

he

rese

arch

are

a to

�3 4�of

its

ori

gin

al s

ize.

If t

he

cen

ter

for

the

redu

ctio

n i

s O

(0,0

),w

hat

are

the

coor

din

ates

of

the

new

res

earc

h a

rea?

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B��2 47 �

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�,D��3 4� ,

��3 4� �

Pra

ctic

e (

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rag

e)

Tran

sfo

rmat

ion

s w

ith

Mat

rice

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-7

9-7



Readin

g t

o L

earn

Math

em

ati

csTr

ansf

orm

atio

ns

wit

h M

atri

ces

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-7

9-7

©G

lenc

oe/M

cGra

w-H

ill51

9G

lenc

oe G

eom

etry

Lesson 9-7

Pre-

Act

ivit

yH

ow c

an m

atri

ces

be

use

d t

o m

ake

mov

ies?

Rea

d th

e in

trod

uct

ion

to

Les

son

9-7

at

the

top

of p

age

506

in y

our

text

book

.

•W

hat

kin

d of

tra

nsf

orm

atio

n s

hou

ld b

e u

sed

to m

ove

a po

lygo

n?

refl

ecti

on

,tra

nsl

atio

n,o

r ro

tati

on

•W

hat

kind

of

tran

sfor

mat

ion

shou

ld b

e us

ed t

o re

size

a p

olyg

on?

dila

tio

n

Rea

din

g t

he

Less

on

1.W

rite

a v

erte

x m

atri

x fo

r ea

ch f

igu

re.

a.�

AB

Cb

.par

alle

logr

am A

BC

D

2.M

atch

eac

h t

ran

sfor

mat

ion

fro

m t

he

firs

t co

lum

n w

ith

th

e co

rres

pon

din

g m

atri

x fr

omth

e se

con

d or

th

ird

colu

mn

.In

eac

h c

ase,

the

vert

ex m

atri

x fo

r th

e pr

eim

age

of a

fig

ure

is m

ult

ipli

ed o

n t

he

left

by

one

of t

he

mat

rice

s be

low

to

obta

in t

he

imag

e of

th

e fi

gure

.A

ll r

otat

ion

s li

sted

are

cou

nte

rclo

ckw

ise

thro

ugh

th

e or

igin

.(S

ome

mat

rice

s m

ay b

e u

sed

mor

e th

an o

nce

or

not

at

all.)

a.re

flec

tion

ove

r th

e y-

axis

vii.

v.b

.90

°ro

tati

onvi

iic.

refl

ecti

on o

ver

the

lin

e y

�x

iiiii

.vi

.d

.27

0°ro

tati

onv

e.re

flec

tion

ove

r th

e or

igin

iiii

i.vi

i.f.

180°

rota

tion

iig.

refl

ecti

on o

ver

the

x-ax

isvi

iiv

.vi

ii.

h.

360°

rota

tion

i

Hel

pin

g Y

ou

Rem

emb

er3.

How

can

you

rem

embe

r or

qu

ickl

y fi

gure

ou

t th

e m

atri

ces

for

the

tran

sfor

mat

ion

s in

Exe

rcis

e 2?

Vis

ual

ize

or

sket

ch h

ow t

he

“un

it p

oin

ts”

(1,0

) o

n t

he

x-ax

is a

nd

(0,1

) o

n t

he

y-ax

is a

re m

oved

by

the

tran

sfo

rmat

ion

.Wri

te t

he

ord

ered

pai

rfo

r th

e im

age

po

ints

in a

2 �

2 m

atri

x,w

ith t

he

coo

rdin

ates

of

the

imag

e o

fth

e x-

axis

un

it p

oin

t in

th

e fir

st c

olu

mn

an

d t

he

coo

rdin

ates

of

the

imag

e o

fth

e y-

axis

un

it p

oin

t in

th

e se

con

d c

olu

mn

.

0�

1 1

0

0�

1 �

10

10

0�

10

1 1

0

�1

0

01

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0

0�

1

0

1 �

10

10

01

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�1

41

0

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03

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1

4�

4

BC

AD

x

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B

C

A

x

y

O

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lenc

oe/M

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w-H

ill52

0G

lenc

oe G

eom

etry

Vec

tor

Add

itio

nV

ecto

rs a

re p

hys

ical

qu

anti

ties

wit

h m

agn

itu

de a

nd

dire

ctio

n.F

orce

an

d ve

loci

ty a

re t

wo

exam

ples

.We

wil

l in

vest

igat

e ad

din

g ve

ctor

qu

anti

ties

.Th

e su

m o

f tw

o ve

ctor

s is

cal

led

are

sult

ant

vect

oror

just

th

e re

sult

ant.

Tw

o se

par

ate

forc

es,o

ne

mea

suri

ng

20 u

nit

s an

d t

he

oth

er m

easu

rin

g 40

un

its,

act

on a

n

obje

ct.I

f th

e an

gle

bet

wee

n t

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es i

s 50

°,fi

nd

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em

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itu

de

and

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ecti

on o

f th

e re

sult

ant

forc

e.

Fir

st,t

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vect

ors

mu

st b

e re

arra

nge

d by

pla

cin

g th

e ta

il o

f th

e 20

-un

it v

ecto

r at

th

e h

ead

of t

he

40-u

nit

vec

tor.

Sin

ce t

hes

e ve

ctor

s ar

e n

ot p

erpe

ndi

cula

r,th

e h

oriz

onta

l an

d ve

rtic

alco

mpo

nen

ts o

f on

e of

th

e ve

ctor

s m

ust

be

fou

nd.

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ng

trig

onom

etry

,th

e h

oriz

onta

l co

mpo

nen

t m

ust

be

(20

cos

50°)

u

nit

s an

d th

e ve

rtic

al c

ompo

nen

t m

ust

be

(20

sin

50°

) u

nit

s.R

epla

cin

g th

e 20

-un

it v

ecto

r w

ith

th

ese

com

pon

ents

,we

can

n

ow f

orm

tw

o ve

ctor

s pe

rpen

dicu

lar

and

use

th

e P

yth

agor

ean

T

heo

rem

to

fin

d th

e re

sult

ant.

r2�

(40

�20

cos

50°

)2�

(20

sin

50°

)2

r2�

(52.

9)2

�(1

5.3)

2

r2�

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.1

tan

O�� 40

2 �02s 0in

c5 os0°50

°�

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2898

m�

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eref

ore,

the

resu

ltan

t fo

rce

is 5

5.1

un

its

dire

cted

16°

from

th

e 40

-un

it f

orce

.

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ve.R

oun

d a

ll a

ngl

e m

easu

res

to t

he

nea

rest

deg

ree.

Rou

nd

all

oth

er m

easu

res

to t

he

nea

rest

ten

th.

1.A

pla

ne

flie

s du

e w

est

at 2

50 k

ilom

eter

s pe

r h

our

wh

ile

the

win

d bl

ows

sou

th a

t 70

kilo

met

ers

per

hou

r.F

ind

the

plan

e’s

resu

ltan

t ve

loci

ty.

259.

6 km

/h,1

6�so

uth

of

wes

t

2.A

pla

ne

flie

s ea

st f

or 2

00 k

m,t

hen

60°

sou

th o

f ea

st f

or 8

0 km

.Fin

d th

e pl

ane’

s di

stan

cean

d di

rect

ion

fro

m i

ts s

tart

ing

poin

t.

249.

8 km

,16�

sou

th o

f ea

st

3.O

ne

forc

e of

100

un

its

acts

on

an

obj

ect.

An

oth

er f

orce

of

80 u

nit

s ac

ts o

n t

he

obje

ct a

t a

40°

angl

e fr

om t

he

firs

t fo

rce.

Fin

d th

e re

sult

ant

forc

e on

th

e ob

ject

.

169.

3 u

nit

s,18

�fr

om

th

e 10

0-u

nit

fo

rce

O

20

40 �

20

cos

50�

20 s

in 5

0�r

20

20 c

os 5

0�

20 s

in 5

0�

50�

O

20 u

nits

40 u

nits

50�

O

20 u

nits

40 u

nits

50�

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-7

9-7

Exam

ple

Exam

ple



Chapter 9 Assessment Answer Key Form 1 Form 2APage 521 Page 522 Page 523

(continued on the next page)

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

A

B

C

A

B

D

D

C

B

D

A

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

C

C

C

B

D

B

A

D

A

120�counterclockwiseor 240� clockwise

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

C

A

D

D

A

B

C

C

D

B


Chapter 9 Assessment Answer KeyForm 2A (continued) Form 2BPage 524 Page 525 Page 526

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

C

B

D

B

D

A

B

A

C

C

��35

�, �45

��

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

D

C

C

C

A

C

B

A

A

C

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

B

B

B

D

A

D

B

A

B

D

��9, �12�


Chapter 9 Assessment Answer KeyForm 2CPage 527 Page 528

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

(5, �2)

1

No, it is not areflection in

line b.

W�(3, �2),X�(�8, 2)

A�(1, 3) B�(5, 1)

(1, �3)

No, each �measure is 150,which is not afactor of 360.

uniform

reduction

x

y

O

A�

B�

C�

A

B

C

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

3

�34

�

203.2�

10

�7, 4�

yes

(�4, 7)

E�(9, �1),F �(2, �5), G�(5, 2)

A�(0, 2), B�(�3, 6),C�(�5, 0)

about 401.1 mph,about 4.3� south

of due west

M�

N�

O�M

N

OC


Chapter 9 Assessment Answer KeyForm 2DPage 529 Page 530

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

(3, 6)

2

Yes, it is reflectedover both lines a

and b.

U �(�1, 0), V �(2, 3)

C �(�4, 0),D�(�4, 3)

Q �(�1, 2)

No, each anglemeasure is 156,which is not afactor of 360.

uniform,semi-regular

enlargement

x

y

OR�

P�

Q�

P

QR

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

35

�43

�

213.7�

13

�9, �6�

yes

(�13, �4)

J �(1, �1), K�(12, 3),L�(4, �8)

D�(�2, �1),E�(�1, �6),F�(3, �2)

about 302.7 mph,about 7.6� east of

due south

C�

D�

E�

C

D

E

G


Chapter 9 Assessment Answer KeyForm 3Page 531 Page 532

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Sample answers:distance measure,

betweenness ofpoints, � measure,

collinearity

C(1, 13), D(�4, 9)

order: 20,magnitude: 18�

R �(3, �5), S �(8, �7),T �(1, �10),

90� clockwise

No, each anglemeasure is 144, whichis not a factor of 360.

not uniform

9

�

AB

C

A� B�

C�

m

x

y

O

V�

T�

U�

T

U

V

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

6.6 in. by 8.4 in.

�65

�

Reflect in the x-axisand the y-axis, in

either order.

333.4�

7.3

�3, 8�

(4 � a, �7 � b)

J �(�2, 5),K �(�8, 1), L�(�2, 3)

G�(5, 4),H�(�1, �1),

I�(�6, 2)

(3, 3�3� )

C


Chapter 9 Assessment Answer KeyPage 533, Open-Ended Assessment

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts ofreflections, translations, rotations, dilations, tessellations,vectors, and matrices.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Figures are accurate and appropriate.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of reflections,translations, rotations, dilations, tessellations, vectors, andmatrices.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Figures are mostly accurate and appropriate.• Satisfies all requirements of problems.

• Shows an understanding of the concepts of reflections,translations, rotations, dilations, tessellations, vectors, andmatrices.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Figures are mostly accurate and appropriate.• Satisfies all requirements of problems.

• Final computation is correct.• No written explanations or work shown to substantiate the

final computation.• Figures may be accurate but lack detail or explanation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of the concepts ofreflections, translations, rotations, dilations, tessellations,vectors, and matrices.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Figures are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer given.

0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation

4 SuperiorA correct solution that is supported by well-developed, accurateexplanations


An

swer

s

Chapter 9 Assessment Answer KeyPage 533, Open-Ended Assessment

Sample Answers

1. Students should draw a tessellationwhere they use only one regular polygonand the angle measures at each vertexare congruent and total 360°, as shownin the figure.

2. A rectangular mirror with two lines ofsymmetry, one vertical and one horizontal,through the middle or a spoon with oneline of symmetry down the middle, etc.

3. a.

b. B�(0, 1)

c. A�(�1, 0)

d. The first column of the matrix in parta is the image of (1, 0) and the secondcolumn is the image of (0, 1).

4. The student should draw a tessellationformed by two different regular polygonsas shown in the figure.

5.

Reflect �ABC with A(1, 1), B(2, 3), andC(�1, 2) in the x-axis, rotate 90°counterclockwise about the origin,translate 2 units to the left and 1 unitup, then dilate with center at the originand a scale factor of 2. The image of�ABC has vertices A�(�2, 4), B�(2, 6),and C�(0, 0).

6. CD� has magnitude �41� or �6.4 anddirection 51.3°. In general, if D hascoordinates (a, b) then the magnitude

will be �a2 � b�2� and the direction will

be tan�1��ab

��.

x

y

O C

D

x

y

O C�

B�

A�

CA

B

0 �11 0

In addition to the scoring rubric found on page A28, the following sample answers may be used as guidance in evaluating open-ended assessment items.


Chapter 9 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 534 Page 535 Page 536

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

isometry

reflection

rotation

standard position

translation

vector

line of symmetry

dilation

parallel vectors

component form

the fixed point around which thepoints are turned

a constant

a pattern covering aplane by transforming

the same figure(s)with no overlapping

or spaces

1.

2.

3.

4.

5.

Q�(6, 4)

1

The points are notall moved the samedistance or in thesame direction.

(x � 9, y � 4)

x

y

O

Q�

P�

R�

1.

2.

3.

4.

5.

A�(�3, �2)

yes

yes

42

P�Q�

R�R

Q

P

T

Quiz 2Page 206 Quiz 4

Page 206

1.

2.

3.

4.

5.

distance

X�(�4, 12), Y�(20, 28),

Z�(8, �16)

magnitude: �17�,direction: 284°

B

x

y

O

D�

F�

E�D

E

F

1.

2.

3.

4.

5.

Z�(�8, 12), Y �(�4, 16)

M�(�7, 3), N�(�5, 2),O�(�6, 6)

A�(3, �1), B�(�8, 2),C �(�5, 7)

D��1, ��32

��, E��2, �12

��,F��1, �

32

��0 �11 0


Chapter 9 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 537 Page 538

An

swer

s

Part I

Part II

6.

7.

8.

9.

10.

(7, 1)

D �(�1, 1), E �(�1, 6),F �(4, 6), G�(4, 1)

No, the size hasbeen changed.

B �(�7, 4)

No; measure ofinterior angle � 168.

1.

2.

3.

4.

5.

B

B

B

C

D

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

m�T � 166 andm�S � 14sometimes

4�2�

F��n2

�, b�

23 � x � 113

18

�45� or 6.7

m�S � 49.6,m�V � 48.4,

v � 39.3

no; B�C� ⁄|| D�A�

a � 10, b � 4,m�HJK � 78

�13, �9�

P�

Q�

R�

x

y

O

P

QR

M


Chapter 9 Assessment Answer KeyStandardized Test Practice

Page 539 Page 541

1.

2.

3.

4.

5.

6.

7. A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D 8. 9.

10. 11.

12.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

13.

14.

15.

m�1 � 77,m�2 � 103

(b � c, a)

E �(7, 6) andF �(�6, �12)

9 6 0 . 1

5 / 7

1 8 6

0 . 8

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Chapter 9 Resource Masters - Math Problem Solving · ©Glencoe/McGraw-Hill iv Glencoe Geometry Teacher’s Guide to Using the Chapter 9 Resource Masters The Fast FileChapter Resource