Content Standards 9-3 G.CO.4 Also G.CO.2, G.COmspachecoephs.weebly.com/.../common_core_rotations.pdf · Lesson 9-3 Rotations 561 9-3 Objective To draw and identify rotation images

Lesson 9-3 Rotations 561

9-3

Objective To draw and identify rotation images of �gures

Rotations

In the diagram, the point (3, 2) is rotated counterclockwise about the origin. The point (x1, y1) is the result of a 90˚ rotation. The point (x2, y2) is the result of a 180˚ rotation, and the point (x3, y3) is the result of a 270˚ rotation. What are the coordinates of (x1, y1), (x2, y2), and (x3, y3)? What do you notice about how the coordinates of the points relate to the coordinates (3, 2) after each rotation?

geom12_se_ccs_c09l03_t06.ai

(3, 2)

(x3, y3 )

(x1, y1 )

(x2, y2 )

x

y

counterclockwise about the origin. The point (x(xpoint (xWhat are the coordinates of (xand (xcoordinates of the points relate to the coordinates coordinates of the points relate to the coordinates (3, 2) after each rotation?(3, 2) after each rotation?

Notice the position of the point, in relation to the x- and y-axis, as it rotates around the origin.

In the Solve It, you thought about how the coordinates of a point change as it turns, or rotates, about the origin on a coordinate grid. In this lesson, you will learn how to recognize and construct rotations of geometric �gures.

Essential Understanding Rotations preserve distance, angle measures, and orientation of �gures.

Lesson Vocabularyrotationcenter of rotationangle of rotation

LessonVocabularyLesson Vocabularyrotation

LessonLesson LessonLesson VocabularyVocabularyVocabularyVocabularyVocabulary

Dynamic ActivityRotations, Reflections, and Translations

AC T I V I T I

E S

Dynamic Activity

AC

AC

AC TC I E

S

DYNAMIC Dynamic Activity

Key Concept Rotation About a Point

A rotation of x about a point Q, called the center of rotation, is a transformation with these two properties:

Q is itself (that is, Q Q).V, QV QV and

m VQV x.

rotates is the angle of rotation.

A rotation about a point is a rigid motion. You write the x rotation of UVW about point Q as r(x , Q)( UVW) U V W .

V

Q

Q

W

U

U

WV

x

V U

WV

The preimage V andits image V areequidistant fromthe center of rotation.

hsm11gmse_0903_t08068.aiUnless stated otherwise, rotations in this book are counterclockwise.

Content StandardsG.CO.4 Develop definitions of rotations . . . in terms of angles, circles, perpendicular lines, parallel lines, and line segments.Also G.CO.2, G.CO.6

MATHEMATICAL PRACTICES

GEOM12_SE_CCS_C09L03.indd 561 7/5/11 10:25:02 AM

CC-11 Rotations 1

CC-11

HSM12GESE_CC11.indd 1 8/1/11 1:26:23 PM

Problem 1

Got It?

562 Chapter 9 Transformations

Drawing a Rotation Image

What is the image of r(100 , C)( LOB)?

Step 1 Draw CO. Use a protractor to draw a 100 angle with vertex C and side CO.

Step 2 Use a compass to construct CO CO.

Step 3 Locate B and L in a similar manner.

Step 4 Draw L O B .

1. Copy LOB from Problem 1. What is the image of LOB for a 50 rotation about B?

When a �gure is rotated 90 , 180 , or 270 about the origin O in a coordinate plane, you can use the following rules.

L

O

C

B

hsm11gmse_0903_t08070.ai

L

O

C

100

B


L

O

C

O

B


L

O

C

B

LO

B

hsm11gmse_0903_t08073.aihsm11gmse_0903_t08074.ai

L

O

C

B

LO

B

Key Concept Rotation in the Coordinate Plane

r(90 , O)(x, y) ( y, x) r(180 , O)(x, y) ( x, y)

r(270 , O)(x, y) (y, x) r(360 , O)(x, y) (x, y)


2 4 62

2

46x

2

4y

O

G (2, 3)G ( 3, 2)


G (2, 3)

G ( 2, 3)

180 2 4 6246

x

2

2

4y


2 4 6

2

246x

2

4y

270

G (2, 3)

G (3, 2)


42 62

2

46x

2

4y

G (2, 3)

360

Step 1Draw protractor to draw

vertex

How do you use the definition of rotation about a point to help you get started?You know that O and O must be equidistant from C and that m OCO must be 100.


2 Common Core


Problem 2

Got It?

Problem 3

Got It?


Drawing Rotations in a Coordinate Plane

PQRS has vertices P(1, 1), Q(3, 3), R(4, 1), and S(3, 0). What is the graph of r(90 , O)(PQRS).

P r(90 , O)(1, 1) ( 1, 1)

Q r(90 , O)(3, 3) ( 3, 3)

R r(90 , O)(4, 1) ( 1, 4)

S r(90 , O)(3, 0) (0, 3)

Next, connect the vertices to graph P Q R S .

2. Graph r(270 , O)(FGHI).

You can use the properties of rotations to solve problems.

Using Properties of Rotations

In the diagram, WXYZ is a parallelogram, and T is the midpoint of the diagonals. How can you use the properties of rotations to show that the lengths of the opposite sides of the parallelogram are equal?

Because T is the midpoint of the diagonals, XT ZT and WT YT. Since W and Y are equidistant from T, and the measure of WTY 180, you know that r(180 , T)(W) Y . Similarly, r(180 , T)(X) Z.

You can rotate every point on WX in this same way, so r(180 , T)(WX) YZ .

Likewise, you can map WZ to YX with r(180 , T)(WZ) YX .

Because rotations are rigid motions and preserve distance, WX YZ and WZ YX .

3. Can you use the properties of rotations to prove that WXYZ is a rhombus? Explain.


2 4 6

2

4 26x

4y

R

SO

P

Q

P ( 1, 1)

Q ( 3, 3)

R ( 1, 4)S (0, 3)


4246

4

2

4y

F ( 3, 2)

I (0, 1)

H ( 1, 1)G ( 3, 1)

x6


W Z

YX

T

What is the graph of

Next, connect the vertices to graph

How do you know where to draw the vertices on the coordinate plane?Use the rules for rotating a point and apply them to each vertex of the figure. Then graph the points and connect them to draw the image.

Got It?

that the lengths of the opposite sides of the parallelogram are equal?

Because Since you know that

You can rotate every point on

Likewise, you can map

Because rotations are rigid motions and preserve distance,

What do you know about rotations that can help you show that opposite sides of the parallelogram are equal?You know that rotations are rigid motions, so if you show that the opposite sides can be mapped to each other, then the side lengths must be equal.


CC-11 Rotations 3


Lesson Check


Practice and Problem-Solving Exercises

Copy each �gure and point P. Draw the image of each �gure for the given rotation about P. Use prime notation to label the vertices of the image.

9. 60 10. 90 11. 180 12. 90

Copy each �gure and point P. �en draw the image of JK for a 180 rotation about P. Use prime notation to label the vertices of the image.

13. 14. 15. 16.

PracticeA See Problem 1.


B

A

D

P


R

P

E

T C


DP

R

B


T

KP


J P K


JP

K


J

K

P


K

J P

Do you know HOW? 1. Copy the �gure and point P. Draw r(70 , P)( ABC).

A

B P

C


In the �gure below, point A is the center of square SQRE.

2. What is r(90 , A)(E)?

3. What is the image of RQ for a 180 rotation about A?

4. Use the properties of rotations to describe how you know that the lengths of the diagonals of the square are equal.

Do you UNDERSTAND? 5. Vocabulary A B C is a rotation image of

ABC about point O. Describe how to �nd the angle of rotation.

6. Error Analysis A classmate drew a 115 rotation of PQR about point P, as shown at the right. Explain and correct your classmate’s error.

7. Compare and Contrast Compare rotating a �gure about a point to re�ecting the �gure across a line. How are the transformations alike? How are they di�erent?

8. Reasoning Point P(x, y) is rotated about the origin by 135 and then by 45 . What are the coordinates of the image of point P? Explain


S Q

E R

A

P P

R

R

Q

Q

115





4 Common Core



For Exercises 17–19, use the graph at the right.

17. Graph r(90 , O)(FGHJ).



20. PRS are P( 3, 2), R(2, 5), and S(0, 0). What are the coordinates of the vertices of r(270 , O)( PRS)?

21. V W X Y has vertices V ( 3, 2), W (5, 1), X (0, 4), and Y ( 2, 0). If r(90 , O)(VWXY) V W X Y , what are the coordinates of VWXY?

22. Ferris Wheel located at the point (30, 0). What are the coordinates of the �rst car after a rotation of 270 about the origin?

For Exercises 23–25, use the diagram at the right. TQNV is a rectangle. M is the midpoint of the diagonals.

23. Use the properties of rotations to show that the measures of both pairs of opposite sides are equal in length.

24. Reasoning Can you use the properties of rotations to show that the measures of the lengths of the diagonals are equal?

25. Reasoning Can you use properties of rotations to conclude that the diagonals of TQNV bisect the angles of TQNV? Explain.

26. In the diagram at the right, M N is the rotation image of MN about point E. Name all pairs of angles and all pairs of segments that have equal measures in the diagram.

27. Language Arts Symbols are used in dictionaries to help users pronounce is called a schwa. It is used in dictionaries to

represent neutral vowel sounds such as a in ago, i in sanity, and u in focus. What transformation maps a to a lowercase e?

Find the angle of rotation about C that maps the black �gure to the blue �gure.

28. 29. 30.

See Problem 2.


446

4

2

4y

O

F (0, 3) J (3, 2)

H (1, 4)

G ( 4, 1)x

6

See Problem 3.


NM

T

V

Q

ApplyB


M

N

E

N’M’


C


C


C


CC-11 Rotations 5



Car 3

Car 18

Car 3

Car 18

31. Think About a Plan London Eye, contains 32 observation cars. Determine the angle of rotation that will bring Car 3 to the position of Car 18.

How do you �nd the angle of rotation that a car travels when it moves one position counterclockwise?How many positions does Car 3 move?

32. Reasoning P, does an x rotation followed by a y rotation give the same image as a y rotation followed by an x rotation? Explain.

33. Writing Describe how a series of rotations can have the same e�ect as a 360 rotation about a point X.

34. Coordinate Geometry Graph A(5, 2). Graph B, the image of A for a 90 rotation about the origin O. Graph C, the image of A for a 180 rotation about O. Graph D, the image of A for a 270 rotation about O. What type of quadrilateral is ABCD? Explain.

Point O is the center of the regular nonagon shown at the right.

35. F to H.

36. Open-Ended Describe a rotation that maps H to C.

37. Error Analysis Your friend says that AB is the image of ED for a 120 rotation about O. What is wrong with your friend’s statement?

In the �gure at the right, the large triangle, the quadrilateral, and the hexagon are regular. Find the image of each point or segment for the given rotation or composition of rotations. (Hint: Adjacent green segments form 30 angles.)

38. r(120 , O)(B) 39. r(270 , O)(L)

40. r(300 , O)(IB) 41. r(60 , O)(E)

42. r(180 , O)(JK) 43. r(240 , O)(G)

44. r(120 , H)(F) 45. r(270 , L)(M)

46. r(180 , O)(I) 47. r(270 , O)(M)

48. Coordinate Geometry Draw LMN with vertices L(2, 1), M(6, 2), and N(4, 2). 90 rotation about the origin and about

each of the points L, M, and N.

49. Reasoning If you are given a �gure and a rotation image of the �gure, how can you �nd the center and angle of rotation?

A B

C

DO

EF

G

H

I



K

J

I

A

B

DG

F E

L

MH CO

ChallengeC


6 Common Core



Mixed Review

BIG has vertices B( 4, 2), I(0, 3), and G(1, 0). Graph BIG and its re�ection image across the given line.

54. the y-axis 55. the x-axis 56. x 4

Find the value of x. Round answers to the nearest tenth.

57. 58.

Get Ready! To prepare for Lesson 9-4, do Exercises 59–61.

59. What are the coordinates of the image of point A( 2, 3) after two 90 rotations about the origin?

60. What are the coordinates of the image of point T(3, 0) after a re�ection across the y-axis followed by a 180 rotation about the origin?

61. H after a 90 rotation about the origin followed by a re�ection across the x-axis is K(3, 2). What are the coordinates of H?

See Lesson 9-2.

See Lessons 8-2 and 8-3.


34

18 mx


30

50 ft

x

See Lessons 9-2 and 9-3.

Standardized Test Prep

50. What is the image of (1, 6) for a 90 counterclockwise rotation about the origin?

(6, 1) ( 1, 6) ( 6, 1) ( 1, 6)

51. aprons like the one shown. If blue ribbon costs $1.50 per foot, what is the cost of ribbon for six aprons?

$15.75 $42.00

$31.50 $63.00

52. In ABC, m A m B 84. Which statement must be true?

BC AC AC BC AB BC BC AB

53. Use the following statement: If two lines are parallel, then the lines do not intersect.a. What are the converse, inverse, and contrapositive of the statement?b. What is the truth value of each statement you wrote in part (a)? If a statement is

false, give a counterexample.

SAT/ACT

hsm11gmse_0903_t14047

18 in.

24 in.

5 in.

5 in.

5 in.

5 in.

ShortResponse


CC-11 Rotations 7


Documents

Content Standards 9-3 G.CO.4 Also G.CO.2, G.COmspachecoephs.weebly.com/.../common_core_rotations.pdf · Lesson 9-3 Rotations 561 9-3 Objective To draw and identify rotation images