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  • MATH HIGH SCHOOL

    SIMILARITY

    EXERCISES

  • Copyright 2015 Pearson Education, Inc. 2

    Copyright 2015 by Pearson Education, Inc. or its affiliates. All Rights Reserved. Printed in the United States of America. Thispublication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise. For information regarding permissions, request forms and the appropriate contacts within the PearsonEducation Global Rights & Permissions department, please visit www.pearsoned.com/permissions/. This work is solely for the use of instructors and administrators for the purpose of teaching courses and assessing student learning. Unauthorized dissemination, publication or sale of the work, in whole or in part (including posting on the internet) will destroy the integrity of the work and is strictly prohibited. PEARSON and ALWAYS LEARNING are exclusive trademarks in the U.S. and/or other countries owned by Pearson Education, Inc. or its affiliates.

    Copyright 2015 Pearson Education, Inc. 2

  • High School: Similarity

    CONTENTS EXERCISES

    EXERCISES

    LESSON 1: LINING UP SIMILAR FIGURES 5

    LESSON 2: VIEWING SCOPES 7

    LESSON 3: DILATIONS 11

    LESSON 4: DILATIONS: SEGMENT LENGTHS 15

    LESSON 5: ANIMATIONS USING DILATION 21

    LESSON 6: DILATIONS: COORDINATE PLANE 25

    LESSON 7: PUTTING IT TOGETHER 29

    LESSON 11: PROPERTIES OF DILATIONS 31

    LESSON 12: TWO DEFINITIONS OF SIMILARITY 35

    LESSON 13: TRIANGLE SIMILARITY 41

    LESSON 14: TRIANGLE PROPORTIONALITY 45

    LESSON 15: PYTHAGOREAN THEOREM 51

    LESSON 16: FLOODLIGHTS AND SHADOWS 57

    Copyright 2015 Pearson Education, Inc. 3

  • High School: Similarity

    Copyright 2015 Pearson Education, Inc. 4

    CONTENTS ANSWERS

    ANSWERS

    LESSON 2: VIEWING SCOPES 63

    LESSON 3: DILATIONS 64

    LESSON 4: DILATIONS: SEGMENT LENGTHS 66

    LESSON 5: ANIMATIONS USING DILATION 68

    LESSON 6: DILATIONS: COORDINATE PLANE 70

    LESSON 7: PUTTING IT TOGETHER 72

    LESSON 11: PROPERTIES OF DILATIONS 73

    LESSON 12: TWO DEFINITIONS OF SIMILARITY 76

    LESSON 13: TRIANGLE SIMILARITY 78

    LESSON 14: TRIANGLE PROPORTIONALITY 79

    LESSON 15: PYTHAGOREAN THEOREM 81

    LESSON 16: FLOODLIGHTS AND SHADOWS 83

  • High School: Similarity

    EXERCISES

    Copyright 2015 Pearson Education, Inc. 5

    EXERCISES

    1. Write what you already know about transformations that produce congruent figures and transformations that produce non-congruent figures.

    Share your work with a classmate. Did you write the same things?

    2. Write your wonderings about similarity.

    Share your wonderings with a classmate. Does your classmate have the same wonderings?

    3. Write a goal stating what you plan to accomplish in this unit. Write your goals for all units in the same place so you can review past goals as you write goals for this unit.

    4. Based on your previous work, write three things you will do differently during this unit to increase your success. Write your strategies for all units in the same place so you can review your past strategies as you write new strategies for this unit.

    For example, consider ways you will participate in classroom discussions, your study habits, how you will organize your time, what you will do when you have a question, and so on.

    LESSON 1: LINING UP SIMILAR FIGURES

  • Copyright 2015 Pearson Education, Inc. 7

    High School: Similarity

    EXERCISESLESSON 2: VIEWING SCOPES

    EXERCISES

    1. You are 60 ft away from a giraffe, and you are viewing it with a scope that is 12 in. long and 4 in. in diameter. The image of the giraffe exactly fills your scope.

    Approximately how tall is the giraffe?

    60 feetNote: Not drawn to scale

    12 in.

    4 in.

    A 10 ft

    B 15 ft

    C 20 ft

    D 25 ft

    2. You are viewing a giraffe through a scope, and the giraffe exactly fills the scope. If you walk backward, then the giraffe should get than your field of vision.

    A smaller

    B larger

  • Copyright 2015 Pearson Education, Inc. 8

    High School: Similarity

    EXERCISESLESSON 2: VIEWING SCOPES

    3. You are viewing a building from 180 ft away through a scope that is 3 in. long. The building is 45 ft tall and exactly fills your scope.

    What is the diameter of your scope?

    180 ftNote: Not drawn to scale

    45 ft

    3 in.

    A 0.5 in.

    B 0.75 in.

    C 1 in.

    D 1.5 in.

    4. Your viewing scope is 8 in. long and 3 in. in diameter. From where you are standing 100 ft away, you cannot see the top of the Ferris wheel you are viewing.

    Describe the possible heights of the Ferris wheel.

    5. Jason, Kayla, and Lian have three different viewing scopes. In terms of distance, in what order should they stand away from an object they are viewing so that it exactly fills each of their scopes?

    A Jason, Kayla, Lian, object

    B Kayla, Lian, Jason, object

    C Jason, Lian, Kayla, object

    D Lian, Jason, Kayla, object

    Jason

    Lian

    Kayla

  • High School: Similarity

    Copyright 2015 Pearson Education, Inc. 9

    EXERCISESLESSON 2: VIEWING SCOPES

    6. Which diagram properly demonstrates a dilation, and only a dilation?

    A

    B

    C

    D

    7. Use a millimeter scale to measure corresponding heights and widths of each letter shown.

    E

    16 point Palatino

    E48 point Palatino

    E72 point Palatino

    Rosa thinks all three letters are mathematically similar. Do your measurements support this conclusion? Explain your answer.

  • High School: Similarity

    Copyright 2015 Pearson Education, Inc. 10

    EXERCISESLESSON 2: VIEWING SCOPES

    Challenge Problem

    8. Sarah is 10 ft away from a wall that she is viewing through a triangular scope.

    What is the area of the part of the wall that can she see?

    2 in.

    3 in.

    5 in.

  • Copyright 2015 Pearson Education, Inc. 11

    High School: Similarity

    EXERCISES

    EXERCISES

    1. The three triangles shown are similar figures. Which point is the center of dilation?

    DC

    B

    A

    A Point A

    B Point B

    C Point C

    D Point D

    2. Sketch the center of this dilation.

    LESSON 3: DILATIONS

  • High School: Similarity

    Copyright 2015 Pearson Education, Inc. 12

    EXERCISES

    3. Suppose you dilate this figure so that the dilation touches point K.

    K

    Center of Dilation

    The dilated image will be than the original figure.

    A larger

    B smaller

    4. If you dilate this triangle, how many pairs of parallel line segments (between the original image and the dilated image) will result?

    5. One of polygons S or T is similar to polygon R. Use measurements to explain which one is similar, and why.

    T

    R

    S

    LESSON 3: DILATIONS

  • High School: Similarity

    Copyright 2015 Pearson Education, Inc. 13

    EXERCISESLESSON 3: DILATIONS

    6. Look at this set of figures. Select the figures in this set that are similar.

    8

    4

    6

    5.6

    60100

    110

    2

    4

    1

    3.3

    11060

    100

    2.8

    4

    2

    3

    110

    60

    100

    A

    7

    3

    5

    4.5

    60100

    110

    D

    B

    1.5

    2.5

    11.4

    110

    60100

    E

    C

    7. Look at this set of figures. Select the figures in this set that are similar.

    3 4

    5

    2.2

    2.5

    2

    6

    10

    8

    BC

    D 4.5

    6

    8

    E

    2

    3

    4

    A

    8. Look at this set of figures. Select the figures in this set that are similar.

    2 6

    6

    84

    4

    2

    1

    3

    3

    9

    4

    A C

    E

    1.5

    2.53.5

    4.5

    D

    1.2

    3.5

    2.5

    4.8B

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    Copyright 2015 Pearson Education, Inc. 14

    EXERCISESLESSON 3: DILATIONS

    9. Select the statements that are true. There may be more than one true statement.

    A All circles are similar to each other.

    B All squares are similar to each other.

    C All rectangles are similar to each other.

    D All regular octagons are similar to each other.

    E All hexagons are similar to each other.

    Challenge Problem

    10. Sketch a dilated image of this figure across the center of dilation C.

    C

  • High School: Similarity

    EXERCISES

    Copyright 2015 Pearson Education, Inc. 15

    LESSON 4: DILATIONS: SEGMENT LENGTHS

    EXERCISES

    1. Draw a dilation of ABC.

    A

    BC

    2. Look at this dilation.

    X

    Z

    YA

    CO

    B

    Which angles are congruent?

    a. A ?A X

    B Y

    C Z

    b. B ?A X

    B Y

    C Z

    c. C ?A X

    B Y

    C Z

  • High School: Similarity

    EXERCISES

    Copyright 2015 Pearson Education, Inc. 16

    3. Look at this dilation.

    X

    Z

    YA

    CO

    B

    The sides of the larger triangle are dilations of which sides of the smaller triangle?

    a. XZ is a dilation of ?

    A AB

    B BC

    C AC

    b. XY is a dilation of ?

    A AB

    B BC

    C AC

    c. YZ is a dilation of ?

    A AB

    B BC

    C AC

    4. What is the center point of this dilation?

    X

    Z

    YA

    CO

    B

    Point

    A A

    B B

    C C

    D O

    LESSON 4: DILATIONS: SEGMENT LENGTHS

  • High School: Similarity

    Copyright 2015 Pearson Education, Inc. 17

    EXERCISESLESSON 4: DILATIONS: SEGMENT LENGTHS

    5. What is the scale factor between similar figures ABCD and A'B'C'D', given the following measurements?

    OA = 4 units OA' = 6.4 units

    OB = 5.5 units OB' = 8.8 units

    OC = 4 units OC' = 6.4 units

    OD = 4.5 units OD' = 7.2 units

    A 1.4

    B 1.6

    C 2.4

    D 3.6

    6. A segment of length 3 units is dilated by a scale factor of 5. What is the resulting length of the new dilated segment?

    A 15 units

    B 8 units

    C 53

    units

    D 35

    units

    7. This diagram shows rectangle ABCD and its dilation A'B'C'D'. What is the scale factor?

    A 3

    B 13

    C 47

    D 3

    A'

    C'

    B'

    A

    O

    C

    B D'

    D

    21

    12

    7

    A

    B

    C

    A'

    B'

    4

    C'

    D'

    D

  • High School: Similarity

    Copyright 2015 Pearson Education, Inc. 18

    EXERCISESLESSON 4: DILATIONS: SEGMENT LENGTHS

    8. How many dilations of ABC are possible with scale factor 4?

    A

    BC

    A 0 dilations

    B 1 dilation

    C 2 dilations

    D An infinite number of dilations

    9. How many dilations of ABC are possible with scale factors between 4 and 6?

    A

    BC

    A 0 dilations

    B 1 dilation

    C 2 dilations

    D An infinite number of dilations

  • High School: Similarity

    Copyright 2015 Pearson Education, Inc. 19

    EXERCISESLESSON 4: DILATIONS: SEGMENT LENGTHS

    10. These two figures are congruent. Is there a center of dilation? Why or why not?

    Challenge Problem

    11. The statement If two figures are congruent, then they are similar is true. However, the converse statement is only true under certain conditions.

    Explain why.

  • High School: Similarity

    EXERCISES

    Copyright 2015 Pearson Education, Inc. 21

    EXERCISES

    1. A dilation of ABC increases each length by 25%. What is the scale factor?

    A 0.25

    B 1.25

    C 25

    D 75

    2. Given the following measurements, what is the scale factor of the dilation?

    PA = 5 units AA' = 7 units

    A 2

    B 2.4

    C 10

    D 12

    3. What are the scale factors of the dilations shown in this diagram, in order from the smallest image to the largest image? (The small black triangle is the original. Do not include this figure as a dilation.)

    A 5, 4, 3, 2

    B 2, 3, 4, 5

    C 12

    13

    14

    15

    , , ,

    D 15

    14

    13

    12

    , , ,

    A'

    B'

    AB

    P

    1

    C

    23

    4

    LESSON 5: ANIMATIONS USING DILATION

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    EXERCISESLESSON 5: ANIMATIONS USING DILATION

    4. Sketch a dilation of the figure DEF by a scale factor of 3 using the given center point C.

    E

    D

    F

    2

    C

    5. A scale plan of a building on a plot of land is shown.

    x'

    4 3 2 1 0inches

    What is the actual length of x if the plan uses a 1" to 40' scale?

    A 12 ft

    B 20 ft

    C 30 ft

    D 40 ft

  • High School: Similarity

    Copyright 2015 Pearson Education, Inc. 23

    EXERCISES

    6. Alonso has a photograph that is 15 cm high and 8 cm wide. He takes it to the photo shop to have the technician make a reduced copy with a height of 9 cm.

    What scale factor should the technician use? Justify your answer.

    7. Alonso has a photograph that is 15 cm high and 8 cm wide. He takes it to the photo shop to have the technician make a reduced copy with a height of 9 cm.

    Alonso would like to place the photograph in a frame with the dimensions 12 cm by 8.8 cm. Will the photograph fit into the frame with an equal width border on each edge? Explain why or why not.

    Challenge Problem

    8. When the scale factor is greater than 1, the image will always be an enlargement of the object. When the scale factor is between 0 and 1, the image will always be a reduction of the object. When the scale factor is 1, the object and the image are congruent.

    Make constructions that show these statements to be true, and explain your reasoning.

    LESSON 5: ANIMATIONS USING DILATION

  • High School: Similarity

    EXERCISES

    Copyright 2015 Pearson Education, Inc. 25

    EXERCISES

    1. MNO is a dilation of DEF.

    D(2, 5)

    F (3, 2)

    N (6, 9)M(3, 7.5)

    E

    O

    0 5 100

    5

    10

    y

    x

    What are the coordinates of the center of dilation?

    ( , )

    2. MNO is a dilation of DEF.

    D(2, 5)

    F (3, 2)

    N (6, 9)M(3, 7.5)

    E

    O

    0 5 100

    5

    10

    y

    x

    What are the coordinates of point O?

    ( , )

    LESSON 6: DILATIONS: COORDINATE PLANE

  • High School: Similarity

    Copyright 2015 Pearson Education, Inc. 26

    EXERCISESLESSON 6: DILATIONS: COORDINATE PLANE

    3. MNO is a dilation of DEF.

    D(2, 5)

    F (3, 2)

    N (6, 9)M(3, 7.5)

    E

    O

    0 5 100

    5

    10

    y

    x

    What are the coordinates of point E?

    ( , )

    4. MNO is a dilation of DEF.

    D(2, 5)

    F (3, 2)

    N (6, 9)M(3, 7.5)

    E

    O

    0 5 100

    5

    10

    y

    x

    Use the coordinates to determine the scale factor of the dilation.

  • High School: Similarity

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    EXERCISESLESSON 6: DILATIONS: COORDINATE PLANE

    5. AB has endpoints (3, 4) and (4, 2). It is dilated by a scale factor of 3, centered at the origin. What are the new endpoints of the resulting dilated line segment?

    A (3, 12) and (4, 6)

    B (6, 7) and (6, 1)

    C (9, 12) and (12, 6)

    D (9, 4) and (12, 2)

    6. Draw the resulting quadrilateral after performing a dilation (centered at the origin) of

    scale factor 1

    4.

    7. Draw the resulting quadrilateral after performing a dilation (centered at the origin) of

    scale factor 1

    4.

    What are the vertices of the new quadrilateral? Use decimals if your answer is not a whole number.

    A' ( , )

    B'( , )

    C' ( , )

    D' ( , )

    A

    B

    C

    D

    0 5 100

    5

    10

    y

    x

    A

    B

    C

    D

    0 5 100

    5

    10

    y

    x

  • High School: Similarity

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    EXERCISESLESSON 6: DILATIONS: COORDINATE PLANE

    8. The scale factor is 2, and the center of dilation is (0, 0). Draw dilations of the points shown.

    x848 2 2 4

    y

    4

    8

    4

    2

    2

    8

    6 6

    6

    6

    Challenge Problem

    9. Suppose a center of dilation at (0, 0) is used to dilate this rectangle.

    a. Choose a scale factor for the dilation that is greater than 1.

    b. How does the dilation affect the perimeter of the rectangle?

    c. How does the dilation affect the area of the rectangle?

    d. What conclusions about ratios can you make about the perimeter and area of similar figures?

    A

    B C

    D

    0 5 100

    5

    10

    y

    x

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    Copyright 2015 Pearson Education, Inc. 29

    EXERCISES

    EXERCISES

    1. Read your Self Check and think about your work in this unit.

    Write three things you have learned.

    Share your work with a classmate. Does your classmate understand what you wrote?

    2. What steps do you follow when you dilate a figure in the coordinate plane?

    Share your work with a classmate. Does your classmate understand what you wrote?

    3. Use your notes from class and your thoughts about the unit to add to your math vocabulary list in your notebook.

    Include the vocabulary word or phrase, a definition, and one or more examples. When appropriate, your example should include a diagram, a picture, or a step-by-step problem-solving approach.

    Word or Phrase Definition Examples

    center of dilation The center of a dilation is the point that remains fixed under the dilation. All other points are moved either toward or away from the center.

    It can be the origin or any other point when the dilation is in the coordinate plane.

    Add these words to your vocabulary list.

    congruent

    similar

    dilation

    scale factor

    4. Review the notes you took during the lessons about similarity and dilations. Add any additional ideas you have about the topics to your notes.

    5. Complete any exercises from this unit you have not finished.

    LESSON 7: PUTTING IT TOGETHER

  • High School: Similarity

    Copyright 2015 Pearson Education, Inc. 31

    EXERCISES

    EXERCISES

    1. When you dilate the point (6, 4) by a scale factor of 4 with a center of dilation at the origin, what is the resulting point?

    A (24, 16) B (2, 0)

    C (10, 8) D (24, 16)

    2. The diagram shows quadrilateral ABCD and its dilation quadrilateral A'B'C'D'. Use the coordinate system to determine the scale factor of the dilation.

    0 5 15100

    5

    15

    10

    y

    x

    A

    B

    C

    D

    A'B'

    C'D'

    3. The diagram shows A and its dilation A'.

    5 5 10 15

    10

    5

    5

    y

    x

    55555

    AA'

    What are the coordinates of the center of dilation?

    ( , )

    A 4 B 14

    C 14

    D 4

    LESSON 11: PROPERTIES OF DILATIONS

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    EXERCISES

    4. The diagram shows A and its dilation A'.

    5 5 10 15

    10

    5

    5

    y

    x

    55555

    AA'

    What is the scale factor of the dilation?

    5. Dilate the line segment by a scale factor of 3, with the center of dilation at the origin.

    x848 2 2 4

    y

    4

    2

    8

    10 6 100 6

    10

    6

    12

    6. A line segment with endpoints at (1, 3) and (2, 0) is dilated by a scale factor of 4 (centered at the origin). Show that the original line and the dilated line have the same slope.

    LESSON 11: PROPERTIES OF DILATIONS

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    EXERCISES

    7. Which line, when dilated from the origin, will result in the same line?

    A

    x848 2 2 4

    y

    4

    8

    4

    2

    2

    8

    10 6 106

    10

    6

    10

    6

    B

    x848 2 2 4

    y

    4

    8

    4

    2

    2

    8

    10 6 106

    10

    6

    10

    6

    C

    x848 2 2 4

    y

    4

    8

    4

    2

    2

    8

    10 6 106

    10

    6

    10

    6

    D

    x848 2 2 4

    y

    4

    8

    4

    2

    2

    8

    10 6 106

    10

    6

    10

    6

    8. Prove that the distance between any point (a, b) and the origin is increased by scale factor k when the point is dilated (from the origin) by scale factor k.

    LESSON 11: PROPERTIES OF DILATIONS

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    EXERCISESLESSON 11: PROPERTIES OF DILATIONS

    9. Dilate right triangle ABC by a scale factor of 3 (from the origin).

    Prove that the new dilated triangle is still a right triangle.

    x50 10

    y

    5

    0

    10

    BC

    A

    10. Suppose DE is dilated by a scale factor of 2 (from the origin).

    Show that the length of the new segment is 2 times the original length.

    x4 4

    y

    4

    DE

    Challenge Problem

    11. What is the ratio of AB to BC in the equilateral triangle shown?

    (AB, DB, and EB are angle bisectors.) (Hint: You can use dilations.)

    EC

    A

    B

    D

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    EXERCISESLESSON 12: TWO DEFINITIONS OF SIMILARITY

    EXERCISES

    1. Which image could be a dilation of this triangle? (Note: Triangles are not drawn to scale.)

    8 cm

    37

    53

    6 cm

    10 cm

    90

    A

    8 cm

    38

    53

    6 cm

    10 cm

    89

    B

    6 cm

    37

    53

    5 cm

    7 cm

    90

    C

    12 cm

    37

    53

    9 cm

    16 cm

    90

    D

    2 cm

    37

    53

    1.5 cm

    2.5 cm

    90

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    EXERCISES

    2. Select the definition of similarity that best fits the figures shown.

    O C

    C'B

    B'A

    A'

    E

    E'

    D

    D'

    A Figure ABCDE is similar to figure A'B'C'D'E' by the transformation definition.

    B Figure ABCDE is similar to figure A'B'C'D'E' by the correspondence definition.

    C The figures are not similar.

    D You don't have enough information to know if figure ABCDE and figure A'B'C'D'E' are similar.

    3. Are these two figures similar? If so, what is the scale factor?

    60

    4575

    11

    5.5

    60 45

    75

    A No

    B Yes; scale factor = 2

    C Yes; scale factor = 5

    D Yes; scale factor = 7

    4. Which statements are true about a quadrilateral and its dilation? There may be more than one true statement.

    A The two quadrilaterals are similar for any scale factor.

    B Only two corresponding sides of the quadrilateral are proportional.

    C Only two angles of the two quadrilaterals are congruent.

    D The two quadrilaterals are congruent if the scale factor is 1.

    E The two quadrilaterals are congruent for any scale factor.

    LESSON 12: TWO DEFINITIONS OF SIMILARITY

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    EXERCISES

    5. Select the definition of similarity that best fits the figures shown.

    BC

    D

    E

    S

    P

    Q

    R

    BCPQ

    CDQR

    DERS

    EBSP

    = = =

    A Figure BCDE is similar to figure PQRS by the transformation definition.

    B Figure BCDE is similar to figure PQRS by the correspondence definition.

    C The figures are not similar.

    D You don't have enough information to know if figure BCDE and figure PQRS are similar.

    6. Select the definition of similarity that best fits the figures shown.

    BC

    D

    10055

    60

    AQ

    R

    S 100

    55

    140

    P

    A Figure ABCD is similar to figure PQRS by the transformation definition.

    B Figure ABCD is similar to figure PQRS by the correspondence definition.

    C The figures are not similar.

    D You don't have enough information to know if figure ABCD and figure PQRS are similar.

    LESSON 12: TWO DEFINITIONS OF SIMILARITY

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    EXERCISESLESSON 12: TWO DEFINITIONS OF SIMILARITY

    7. Write one way you could prove that these two triangles are similar.

    x5 5

    y

    5

    5

    10 10

    10

    10

    B (3, 5)A (3, 4)

    A' (6, 8)

    B' (6, 10)

    8. ABC has vertices at (2, 3), (4, 2), and (1, 3). ABC is dilated by a scale factor of 2 to form A'B'C'. Show that A is congruent to A ' .

    x5 5

    y

    5

    5

    10 10

    10

    10

    A' (4, 6)

    B' (8, 4)

    C' (2, 6)

    B (4, 2)

    C (1, 3)

    A (2, 3)

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    EXERCISESLESSON 12: TWO DEFINITIONS OF SIMILARITY

    Challenge Problem

    9. If the scale factor between the two similar triangles shown is x, what is the area of the larger triangle in terms of A, B, and x?

    A

    B

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    EXERCISESLESSON 13: TRIANGLE SIMILARITY

    EXERCISES

    1. Which statement is true about the two triangles shown?

    55

    85

    55

    85

    A The two triangles are similar, but it is uncertain if they are congruent.

    B The two triangles are congruent.

    C It is uncertain if the two triangles are similar from the given information.

    D The two triangles are congruent, but not similar.

    2. Which statement is true about the two triangles shown? (Note: Drawing may not be to scale.)

    8 cm

    6 cm

    3 cm 16 cm

    A The two triangles are similar, but it is uncertain if they are congruent.

    B The two triangles are congruent.

    C It is uncertain if the two triangles are similar from the given information.

    D The two triangles are congruent, but not similar.

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    EXERCISES

    3. These two triangles are similar (shown by a dilation). What is the measure of angle ?

    130

    28

    A 22

    B 28

    C 50

    D 152

    4. These two triangles are similar. Determine the length of the missing side length S.

    6 in.

    6 in.

    S

    9 in.

    4 in.

    12 in.

    A 2 in.

    B 4 in.

    C 6 in.

    D 8 in.

    LESSON 13: TRIANGLE SIMILARITY

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    EXERCISES

    5. Determine whether these triangles are similar. If they are similar, what is the scale factor that can be used to dilate ABC to make EFG?

    A

    EF

    G

    103103

    5 cm

    7 cm

    36

    36

    C

    B

    A ABC is dilated to EFG by a scale factor of 57

    B ABC is dilated to EFG by a scale factor of 1.4.

    C ABC is dilated to EFG by a scale factor of 7.

    D ABC is not similar to EFG.

    6. A right triangle is dilated by a scale factor of 3 with the center of dilation at (0, 2).

    0 5 100

    5

    10

    y

    x

    Which statements are true about the triangle after the dilation? There may be more than one true statement.

    A The side lengths increase by a scale factor of 3.

    B The angle measures increase by a scale factor of 3.

    C All the angle measures will stay the same.

    D The angle measures change differently depending on the distance away from the center of dilation.

    E The side lengths change differently depending on the distance away from the center of dilation.

    LESSON 13: TRIANGLE SIMILARITY

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    EXERCISESLESSON 13: TRIANGLE SIMILARITY

    7. If two triangles have at least congruent angle pairs, then you know the two triangles are .

    A 1, congruent

    B 2, congruent

    C 3, congruent

    D 1, similar

    E 2, similar

    F 3, similar

    Challenge Problem

    8. Examine the information given about the two rhombuses shown. Determine whether you can be certain that they are similar.

    Justify your proof with a sketch.

    6 cm4 cm

    130130

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    EXERCISES

    EXERCISES

    1. In the figure, DE is parallel to AC. The given side lengths are AC = 20 units, DE = 15 units, BD = 9 units, DA = 3 units, EC = 4 units.

    Determine the length of BE.

    43E

    15

    9

    20

    D

    A C

    B

    A BE = 8 units

    B BE = 12 units

    C BE = 16 units

    D BE = 24 units

    2. In the figure, DE is parallel to AB . The given side lengths are AB = 15 units, AD = 12 units, CD = 6 units.

    Determine the length of DE .

    6

    E

    15

    12

    D

    A

    C

    B

    A DE = 3 units

    B DE = 5 units

    C DE = 7.5 units

    D DE = 10 units

    LESSON 14: TRIANGLE PROPORTIONALITY

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    EXERCISES

    3. In the figure, DE is parallel to AC . The given lengths are AC = 50 units, DE = 40 units, and AB is 65 units.

    Determine the lengths of AD and DB .

    65

    E

    50 40

    D

    A

    CB

    A AD = 13 units and DB = 52 units

    B AD = 10 units and DB = 55 units

    C AD = 13 units and DB = 65 units

    D AD = 25 units and DB = 40 units

    4. Determine whether or not segment DE is parallel with segment BC . Justify your response using triangle proportionality theorems.

    4

    3

    E

    20

    6

    10

    D

    A

    C

    B

    30

    LESSON 14: TRIANGLE PROPORTIONALITY

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    EXERCISES

    5. Determine the missing segment lengths DA and EA . Justify your response using triangle proportionality theorems.

    6

    E10

    7D

    A

    C

    B

    8

    6. Prove that segment FG is parallel to side BC . Justify your proof showing two similar triangles.

    4

    5

    F

    12

    8

    27

    G

    A

    C

    B

    10

    LESSON 14: TRIANGLE PROPORTIONALITY

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    EXERCISESLESSON 14: TRIANGLE PROPORTIONALITY

    7. ADE is a dilation of ABC with a center of dilation at point A. Use the AA criterion to explain why ABC and ADE are similar triangles.

    E

    DA

    C

    B

    8. Lines m and n are parallel. Write a proof to show that ABC is similar to DEC.

    DE

    C

    BA

    Line m

    Line n

    9. Decide which two triangles are similar. Explain your reasoning. (Note: The figures are not drawn to scale.)

    9595

    3727

    48

    48A

    B

    C

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    EXERCISESLESSON 14: TRIANGLE PROPORTIONALITY

    10. Decide which two triangles are similar. Explain your reasoning. (Note: The figures are not drawn to scale.)

    7

    9

    26.4

    8

    12

    17.6

    A

    B

    C

    Challenge Problem

    11. ABE and CDE are formed by the diagonals of trapezoid ABCD.

    2.1

    2.7

    3

    5

    A B

    C

    E

    D

    p

    q

    a. These two triangles are similar. Explain why.

    b. Use the similarity ratio to determine p and q, the measures of EB and ED .

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    EXERCISES

    EXERCISES

    1. Rosa was working on her Pythagorean Theorem proof and noticed something interesting. No matter how she manipulated the dimensions of the right triangle, the area of the square ABJK and the area of another figure were always equal.

    Which other figure in the diagram is she referring to?

    B

    A

    E

    O

    CJ

    K

    L

    M

    N

    D

    A Rectangle ADEL

    B Rectangle DCME

    C Square BCNO

    D Triangle BDC

    LESSON 15: PYTHAGOREAN THEOREM

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    EXERCISES

    2. Evan decided to try a different approach for his Pythagorean Theorem proof. He started with four congruent right triangles, all rotations of each other and

    each with an area of ab2

    square units.

    a

    a

    a

    ac

    b

    cb c

    b

    bc

    He put the triangles together in this formation, with the hypotenuse of each triangle forming an outside edge of a large square.

    His idea was to relate the large square, with area c2, to the area of the four triangles, and the area of the small square in the center.

    How could he express the area of the small square in the center?

    A (a b)2

    B (b a)2

    C b2

    D a2

    LESSON 15: PYTHAGOREAN THEOREM

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    EXERCISES

    3. Finish Evans proof. Set up an equation that compares:

    The area of the large square, given as c2

    The area of the four triangles, with each triangle's area given as ab2

    The area of the small square

    You must determine the expression for the area of the small square. Then solve your equation, and explain how it proves the Pythagorean Theorem.

    LESSON 15: PYTHAGOREAN THEOREM

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    EXERCISES

    4. Kayla has another idea to prove the Pythagorean Theorem. She starts by drawing two squares next to each other, with side lengths a and b. The total area of the figure is a2 + b2.

    b

    a

    Next, she draws in two hypotenuses, to make two right triangles with sides a, b, and c. The remaining area is an irregular figure.

    cc

    b

    a

    b a

    Her plan is to rotate the right triangles, so they form a new square with side length c.

    Sketch a diagram showing this rotation and explain how Kayla's rotation diagram proves the Pythagorean Theorem.

    LESSON 15: PYTHAGOREAN THEOREM

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    EXERCISESLESSON 15: PYTHAGOREAN THEOREM

    5. Use the Pythagorean Theorem to determine whether the SSS Similarity Theorem applies to these two triangles. Which statements are true? There may be more than one true statement.

    4

    6

    2

    Triangle BTriangle A

    3

    A The hypotenuse of A is 13 units.

    B The hypotenuse of B is 4 13 units.

    C A is similar to B by a scale factor of 14 .

    D B is similar to A by a scale factor of 2.

    E The triangles are not similar.

    6. Use the Pythagorean Theorem to determine whether the SSS Similarity Theorem applies to these two triangles. Which statements are true? There may be more than one true statement.

    15

    25

    6

    Triangle DTriangle C

    10

    A The second leg of C is 7 units.

    B The second leg of C is 20 units.

    C C is similar to D by a scale factor of 25 .

    D D is similar to C by a scale factor of 3.

    E The triangles are not similar.

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    EXERCISESLESSON 15: PYTHAGOREAN THEOREM

    7. What are the three lengths of a similar triangle that is twice as large as this right triangle?

    6.52.5

    8. Decide which of these triangles are similar: ABC, ADB, and BDC. Explain your reasoning.

    A

    B C

    D

    Challenge Problem

    9. In this diagram, point A is at (0, 12), point B is at (9, 0), point C is at (5, 0), and point D is at (0, 0). Find as many of the side lengths as you can in the figure.

    A

    B C x

    y

    O

    D

    FE

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    EXERCISESLESSON 16: FLOODLIGHTS AND SHADOWS

    EXERCISES

    1. Read your Self Check and think about your work in this unit.

    Write three things you have learned.

    Share your work with a classmate.

    Does your classmate understand what you wrote?

    2. Lian started making a dilation chart with examples for different scale factors.

    Complete her work by adding other examples. Identify all parallel lines.

    Use a center that is not the origin for at least one of the dilation examples.

    Scale Factor Example k Center of Dilation

    k > 1

    0 50

    5

    10y

    x

    A

    B

    C'

    A'

    C

    B'

    1.5 (0, 0)

    0 < k < 1

    k < 1

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    EXERCISESLESSON 16: FLOODLIGHTS AND SHADOWS

    3. Complete this chart to explain the properties of dilations and similarity theorems from the unit.

    Property or Theorem (Lesson Number)

    DefinitionExplain the Property or Theorem Using Your Own Words or Diagrams

    Parallels Under Dilation Property (Lesson 11)

    A dilation takes a line L to a line L', and line L' is parallel to line L (for any center of dilation not on L).

    Segment Lengths Under Dilations Property (Lesson 11)

    A dilation with scale factor k takes a segment with length L to a segment with length |k|L.

    Angle Measures Under Dilations Property (Lesson 11)

    Any dilation takes an angle to an angle of the same measure.

    Correspondence Definition of Similarity (Lesson 12)

    Two polygons are similar if all corresponding angles are congruent and all corresponding side lengths are proportional.

    Dilation Definition of Similarity (Lesson 12)

    Two figures are similar if one can be transformed into the other by a dilation and, if necessary, a sequence of rigid motions.

    AA Triangle Similarity Theorem (Lesson 13)

    If two triangles have two pairs of congruent angles, the triangles must be similar.

    SSS Triangle Similarity Theorem (Lesson 13)

    If two triangles have three pairs of proportional side lengths, the triangles must be similar.

    Triangle Proportionality Theorem (Lesson 14)

    A line parallel to one side of a triangle divides the other two sides proportionally.

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    EXERCISESLESSON 16: FLOODLIGHTS AND SHADOWS

    4. Review the notes you took during the lessons about similarity and dilations. Add any additional ideas you have about the topics to your notes.

    5. Complete any exercises from this unit you have not finished.

  • MATH HIGH SCHOOL

    SIMILARITY

    EXERCISESANSWERS FOR EXERCISES

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    ANSWERSLESSON 2: VIEWING SCOPES

    ANSWERS

    G.SRT.1.b 1. C 20 ft

    G.SRT.1.b 2. A smaller

    G.SRT.1.b 3. B 0.75 in.

    G.SRT.1.b 4. H > 37.5 ft

    The Ferris wheel is greater than 37.5 ft tall.

    G.SRT.1.a 5. C Jason, Lian, Kayla, object

    G.SRT.1.a 6. A

    G.SRT.1.b 7. Rosa is correct because the ratio width : height = 3 : 4 in each letter.

    Challenge Problem

    G.SRT.1.a G.SRT.1.b

    8. You can set up a proportion with the base and the height of the triangle.

    52

    104

    53

    106

    6 42

    in. in.

    ft ft

    in. in.

    ft ft

    ft ft

    = =

    = =

    bb

    hh

    == 12 ft2

    Sarah can see 12 ft2 of the wall.

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    ANSWERSLESSON 3: DILATIONS

    ANSWERS

    G.SRT.1.a 1. C Point C

    G.SRT.1.a 2. C

    G.SRT.1.a 3. A larger

    G.SRT.1.a 4. Every line segment of the original figure will be taken to a parallel line segment. A triangle has three sides, so there will be three pairs of parallel line segments.

    G.SRT.2 5. All corresponding angles are equal in all three figures.

    All edges of polygon T are larger than the corresponding edges of polygon R by a factor of 3. Thus, polygon T is similar to polygon R by a scale factor of 3.

    The edges of the concave section of polygons S and R are in a different ratio than the outer edges. Thus, polygons S and R are not similar.

    G.SRT.2 6. Figure A, Figure C

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    ANSWERSLESSON 3: DILATIONS

    G.SRT.2 7. Figure B, Figure D

    G.SRT.2 8. Figure A, Figure E

    G.SRT.2 9. A All circles are similar to each other.

    B All squares are similar to each other.

    D All regular octagons are similar to each other.

    Challenge Problem

    G.SRT.1.a 10. The dilated image occurs across the center of dilation, so the result will be a 180 rotation (or a composition of two reflections), and then any degree of dilation.

    An example of a dilated image is shown.

    C

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    ANSWERSLESSON 4: DILATIONS: SEGMENT LENGTHS

    ANSWERS

    G.SRT.1.a 1. Here is an example of a dilation.

    A

    BC

    G.SRT.1.a 2. a. A X

    b. B Y

    c. C Z

    G.SRT.1.a 3. a. C AC

    b. A AB

    c. B BC

    G.SRT.1.a 4. D O

    G.SRT.1.b 5. B 1.6

    G.SRT.1.b 6. A 15 units

    G.SRT.1.b 7. D 3

    G.SRT.1.b 8. B 1 dilation

    G.SRT.1.b 9. D An infinite number of dilations

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    ANSWERSLESSON 4: DILATIONS: SEGMENT LENGTHS

    G.SRT.1.a 10. If you draw lines connecting corresponding vertices, the lines are parallel. Thus, there is no center of dilation.

    Challenge Problem

    G.SRT.1.a 11. If two figures are congruent, then they are similar with a scale factor of 1. The converse statement, If two figures are similar, then they are congruent, is only true for a scale factor of 1.

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    ANSWERSLESSON 5: ANIMATIONS USING DILATION

    ANSWERS

    G.SRT.1.b 1. B 1.25

    G.SRT.1.b 2. B 2.4

    G.SRT.1.b 3. B 2, 3, 4, 5

    G.SRT.1.b 4.

    E

    D

    FD'

    F'

    E'

    2

    C

    6

    G.SRT.1.b 5. C 30 ft

    G.SRT.1.b 6.

    If you compare the size the photo was with the size it is reduced to, you can determine the scale factor.

    The scale factor is 35

    , or 0.6.

    G.SRT.1.b 7. Using the scale factor of the width of the reduced photo is

    Borders top and bottom are (12 9) 2 = 1.5 cm. Borders left and right are (8.8 4.8) 2 = 2 cm.

    No, the borders are not an equal width all the way around the photo.

    15 9 5 3 135

    : : := =

    35

    835

    245

    4 8 .= = cm

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    ANSWERSLESSON 5: ANIMATIONS USING DILATION

    Challenge Problem

    G.SRT.1.b 8.

    O

    A

    A' BB'

    CC'

    D

    Enlargement

    Reduction

    D'

    A''

    B''

    C ''

    D''

    The original figure is in the middle.

    The construction to the left uses a scale factor of 0.5, which is between 0 and 1. You can see that A'B'C'D' is smaller than the original image.

    The construction to the right uses a scale factor of 2, which is greater than 1. You can see that A''B''C''D'' is larger than the original image.

    If the scale factor is 1, then the objects are congruent.

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    ANSWERSLESSON 6: DILATIONS: COORDINATE PLANE

    ANSWERS

    G.SRT.1.a 1. (0, 0)

    G.SRT.1.a 2. (4.5, 3)

    G.SRT.1.a 3. (4, 6)

    G.SRT.1.b 4. The scale factor is 1.5, or .

    G.SRT.1.b 5. C (9, 12) and (12, 6)

    G.SRT.1.a 6.

    A

    B

    C

    D

    A'B'

    C'

    D'

    0 5 100

    5

    10

    y

    x

    G.SRT.1.b 7. A' (1, 1.5)

    B' (1.5, 2)

    C' (3, 3)

    D' (2.5, 1)

    32

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    ANSWERSLESSON 6: DILATIONS: COORDINATE PLANE

    G.SRT.1.b 8. The arrows lead to the dilated points.

    x848 2 2 4

    y

    4

    8

    4

    2

    2

    8

    6 6

    6

    6

    Challenge Problem

    G.SRT.1.b 9. a. If the scale factor is 2, then the vertices of the dilated rectangle are (6, 10), (6, 16), (14, 10), and (14, 16).

    b. The perimeter of the original figure is 3 2 + 4 2 = 14 units. The perimeter of the dilated figure is 6 2 + 8 2 = 28 units.

    c. The area of the original figure is 3 4 = 12 square units. The area of the dilated figure is 6 8 = 48 square units.

    d. The ratio of the original perimeter to the dilated perimeter is the same as the scale factor. In this example, you multiply the original perimeter by 2. For area, the ratio is the square of the scale factor. In this example, you multiply the area of the original figure by the square of 2, or 4.

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    ANSWERS

    ANSWERS

    G.SRT.1.a G.SRT.1.b

    3.Word or Phrase Definition Examples

    center of dilationc The center of a dilation is the point that remains fixed under the dilation. All other points are moved either toward or away from the center.

    It can be the origin or any other point when the dilation is in the coordinate plane.

    congruent Two geometric figures are congruent if they have exactly the same size and shape. 2 2

    3

    34

    4

    similar Two figures are similar if they have exactly the same shape, but (possibly) different sizes.

    dilation A dilation is a type of similarity transformation in which all lengths are scaled uniformly.

    scale factor The scale factor is the ratio of two corresponding lengths in the original figure and the dilated figure.

    If the scale factor (k ) of 2 is used, the segments and perimeter of the transformed figure are twice as long as those of the original figure.

    LESSON 7: PUTTING IT TOGETHER

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    ANSWERSLESSON 11: PROPERTIES OF DILATIONS

    ANSWERS

    G.SRT.1.b 1. D (24, 16)

    G.SRT.1.b 2. B 14

    G.SRT.1.b 3. ( 12 , 2 )

    G.SRT.1.b 4. 3.5 or 3 12

    G.SRT.1.b 5.

    x848 2 2 4

    y

    4

    2

    8

    10 6 100 6

    10

    6

    12

    G.SRT.1.b 6. The original slope is 1, and dilated segment has endpoints are (8, 0) and (4, 12) and a slope of 1.

    x848 2 2 4

    y

    4

    2

    8

    10 6 100 6

    10

    6

    12

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    ANSWERSLESSON 11: PROPERTIES OF DILATIONS

    G.SRT.1.b 7. A

    x848 2 2 4

    y

    4

    8

    4

    2

    2

    8

    10 6 106

    10

    6

    10

    6

    G.SRT.1.b 8.By the distance formula, the distance from the point (a, b) to the origin is a b2 2+ .

    The distance between dilated point (ka, kb) and the origin is

    ka kb k a b k a b( ) ( )2 2 2 2 2 2 2+ = + = +( ) . Thus, the distance is increased by a scale factor of k.

    G.SRT.1.b 9.

    x5 100 15

    y

    5

    10

    0

    15

    20

    20

    BC

    A

    B'

    C'

    A'

    You know that the dilated image of a line segment has the same slope as the original line segment, so you know that A B' ' has the same slope as AB and that B C' ' has the same slope as BC. AB is perpendicular to BC, so A B' ' must be perpendicular to B C' '. Thus, the dilated triangle is also a right triangle.

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    ANSWERSLESSON 11: PROPERTIES OF DILATIONS

    G.SRT.1.b 10. The original segment length is determined by D (3, 2) and E (2, 1).

    DE = +

    =

    1 2 2 3

    26

    2 2( ) ( )

    The dilated line segment is determined by D' (6, 4) and E' (4, 2).

    D E' ' = +

    =

    =

    =

    2 4 4 6

    104

    4 26

    2 26

    2 2( ) ( )

    The new dilated length is exactly double the original length.

    Challenge Problem

    G.SRT.1.b 11. Dilate the triangle by a scale factor of 2, with point A as the center of the dilation.

    EC

    A

    B

    D

    FG

    H

    You know that the scale factor creating this dilation is 2. Thus, the distance AH to the center of the dilated triangle is twice the distance AB to the center of the original triangle; in other words, AB = BH. BDis parallel to EH, and BE is parallel to DH by properties of dilation, so figure BEHD is a parallelogram. BH is perpendicular to DE(given), so figure BEHD is a rhombus. Therefore, BC = CH (properties of a rhombus) and BC is half of AB. Thus, the ratio of AB to BC is 2.

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    ANSWERSLESSON 12: TWO DEFINITIONS OF SIMILARITY

    ANSWERS

    G.SRT.1.b 1. D

    2 cm

    37

    53

    1.5 cm

    2.5 cm

    90

    G.SRT.2 2. A Figure ABCDE is similar to figure A'B'C'D'E' by the transformation definition.

    G.SRT.1.b 3. B Yes; scale factor = 2

    G.SRT.2 4. A The two quadrilaterals are similar for any scale factor.

    D The two quadrilaterals are congruent if the scale factor is 1.

    G.SRT.2 5. B Figure BCDE is similar to figure PQRS by the correspondence definition.

    G.SRT.2 6. C The figures are not similar.

    G.SRT.1.b 7. If you prove the two triangles are a dilation of one another, then the triangles are similar. Call the origin point O. The side lengths of the smaller triangle are:

    OA

    OB

    AB

    = ( )

    =

    = ( ) ( )

    3 4 25 5

    3 5 34

    3 3 4 5 36 1 37

    2 2

    2 2

    2 2

    + = =

    + =

    + = + =

    The side lengths of the larger triangle are:

    OA

    OB

    A B

    + = =

    + = =

    + =

    = ( )

    =

    = ( ) ( )

    6 8 100 10

    6 10 136 2 34

    6 6 8 10

    2 2

    2 2

    2 2 1148 2 37=

    OA' = 2(OA), OB' = 2(OB), and A'B' = 2(AB). Thus, the sides are proportional and are dilated by a scale factor of 2.

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    ANSWERSLESSON 12: TWO DEFINITIONS OF SIMILARITY

    G.SRT.2 8. Point A' translates to point A by moving down 3 units and to the right 2 units. Apply this same translation to point C'. The new point is (0, 9), which lies on AC . Then apply this translation to point B', getting point (10, 1), which lies on AB .

    Thus, A must be congruent to A .

    Challenge Problem

    G.SRT.2 9. Since the scale factor is x, multiply the lengths of the smaller triangle by x to find the lengths of the larger triangle.

    Use the Pythagorean Theorem to find the length of the third side of the smaller triangle (call it C): A C B

    C B A

    2 2 2

    2 2

    + =

    =

    Thus, the third side of the larger triangle has a length of x B A2 2 .

    The area of a triangle is (base height), so the area of the larger triangle is:

    Ax x B A Ax B A( ) ( )2 2 2 2 22 2

    =

    x5 5

    y

    5

    5

    10 10

    10

    10

    B (4, 2)

    C (1, 3)

    A (2, 3)

    A

    B

    Ax

    Bx

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    ANSWERSLESSON 13: TRIANGLE SIMILARITY

    ANSWERS

    G.SRT.2 1. A The two triangles are similar, but it is uncertain if they are congruent.

    G.SRT.2 2. C It is uncertain if the two triangles are similar from the given information.

    G.SRT.5 3. A 22

    G.SRT.5 4. D 8 in.

    G.SRT.2 G.SRT.5

    5. B ABC is dilated to EFG by a scale factor of 1.4.

    G.SRT.2 6. A The side lengths increase by a scale factor of 3.

    C All the angle measures will stay the same.

    G.SRT.3 7. E 2, similar

    Challenge Problem

    G.SRT.2 8. Since it is given that the figures are rhombuses, you can label each side length. All four side lengths of the large figure are 6 cm, and all four side lengths of the small figure are 4 cm. Rhombuses can be cut into two isosceles triangles (shown with the dashed line in the sketch). You know that the base angles of all four isosceles triangles (two triangles from each rhombus) measure 65. Knowing this, you can be sure that all of the isosceles triangles are similar, since they all have the same angle measures (65-65-50). Therefore, the two rhombuses are similar. Further, you can determine

    that the scale factor of the dilation from the large figure to the small figure is 23

    , since the side lengths are proportional.

    6 cm4 cm

    65 6565

    6 cm4 cm

    6 cm4 cm

    6 cm4 cm

    65 6565

    65

    65

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    ANSWERSLESSON 14: TRIANGLE PROPORTIONALITY

    ANSWERS

    G.SRT.5 1. B BE = 12 units

    G.SRT.5 2. B DE = 5 units

    G.SRT.5 3. A AD = 13 units and DB = 52 units

    G.SRT.5 4. DE is not parallel to BC. The lengths shown for DE and BC are 20 units and 30 units, indicating that if the segments were parallel, DE would divide the other two triangle

    sides into two sections of 23

    . This is true for side AB, which has a total length of 9

    units and is cut into two lengths of 6 units and 3 units. But this is not true for side AC, which has a total length of 14 units but is cut into two lengths of 10 units and 4 units.

    Since DE does not divide the triangle sides proportionally, you can conclude that DE is not parallel to BC.

    G.SRT.5 5. Since DE is parallel to BC, you know the other two sides AB and AC are divided proportionally. DE = 6 units and BC = 10 units, so the ratios of the other triangle

    sides are 610

    and 410

    .

    BD = 7 units and must equal 410

    of the total length of side AB. To determine length

    AD, calculate 764

    10 5 .= .

    Double check: the total length AB is now 17.5 units. So, 7

    17 5410.

    = and 10 517 5

    610

    .

    .= .

    The proportion works.

    Use similar process for side AC. CE = 8 units and must be 410

    the total length of side AC.

    To determine length AE, calculate 864

    12 = .

    Double check: the total length AC is 20 units. 820

    410

    = and 1220

    610

    = .

    The proportion works. DA = 10.5 units and AE = 12 units.

    G.SRT.2 G.SRT.4

    6. The two similar triangles are AFG and ABC. You can show that these two triangles are similar by showing that all three corresponding side pairs are proportional. The side lengths of AFG are 4 units, 8 units, and 12 units. The side lengths of ABC are 9 units, 18 units, and 27 units. All three corresponding sides follow the same ratio: 49

    818

    1227

    = = . Therefore, the triangles are similar.

    Since the triangles are similar, all three corresponding angle pairs must be congruent. Therefore, you can say that CBF GFA , which forms two lines that both meet the transversal AB at the same angle. Thus BC must be parallel to FG.

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    ANSWERSLESSON 14: TRIANGLE PROPORTIONALITY

    G.SRT.3 7. Given that ADE is a dilation of ABC, .

    Only two of these three statements are needed in the explanation: Therefore, ABC ADE by alternate interior angles. Therefore, ACB AED by alternate interior angles. A A by same angle.

    With two congruent angles, the two triangles are similar by the AA criterion.

    G.SRT.3 8. You only need to prove that two pairs of angles are congruent to show that ABC is similar to DEC.

    ACB and DCE: AD and BE intersect at point C. This intersection forms a set of vertical angles, which are congruent.

    ABC and DEC: BE is a transversal that crosses the set of parallel lines (lines m and n). The intersection forms opposite interior angles, which are congruent.

    ABC and DEC: AD is a transversal that crosses the set of parallel lines (lines m and n). The intersection forms opposite interior angles, which are congruent.

    DE

    C

    BA

    Line m

    Line n

    G.SRT.5 9. If you calculate the third angle of A you set up the equation 95 + 48 + x = 180, or x = 37. Thus, the angles of A are equal to the angles of C. Therefore, A is similar to (AA similarity criteria).

    G.SRT.5 10. All three triangles are isosceles, so if two sides are in the same ratio, then all three sides will be. B is similar to A (SSS similarity criteria with k = 2.2).

    Challenge Problem

    G.SRT.3 G.SRT.5

    11. a. ABE and CDE meet the AA criterion for similarity. In the two triangles, AEB CED because they are vertical angles. BAE EDC and because they are alternate interior angles formed by parallel lines.

    b. The similarity ratio AB : DC = 3 : 5.

    p q= = = =

    35

    2 7 1 653

    2 1 3 5 . . . . and

    So, EB = 1.6 units and ED = 3.5 units.

    DE BC

    ABE ECD

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    ANSWERS

    ANSWERS

    G.SRT.4 1. A Rectangle ADEL

    G.SRT.4 2. B (b a)2

    G.SRT.4 3. The total area of the large square is c2, which is made up of the four right triangles and the small square in the center. You know the area of the small square in the

    center is (b a)2, and each right triangle has area ab2

    . You can set up an equation

    that relates all of the areas and then manipulate the equation to reveal the Pythagorean Theorem.

    c b aab

    b ab a ab

    b a

    2 2

    2 2

    2 2

    42

    2 2

    = +

    = + +

    = +

    ( )

    This demonstrates the Pythagorean Theorem.

    G.SRT.4 4. Here is a sketch showing the two rotations that change the figure into a new square with area c2.

    cc

    b

    a

    b a

    The figure begins with area a2 + b2. Then, without adding or taking away any area, the figure can be manipulated to cover an exact area of c2. This shows that a2 + b2 = c2.

    G.SRT.5 5. A The hypotenuse of A is units.

    D B is similar to A by a scale factor of 2.

    G.SRT.5 6. B The second leg of C is 20 units.

    C C is similar to D by a scale factor of 25 .

    13

    LESSON 15: PYTHAGOREAN THEOREM

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    ANSWERS

    G.SRT.5 7. The leg lengths are 5 units and 12 units, and the hypotenuse length is 13 units.

    G.SRT.4 G.SRT.5

    8. ABC, ADB, and BDC are all right triangles, but one other angle pair needs to be congruent in order to prove similarity using the AA Similarity Theorem.

    BAC and BAD are the same angle, so ABC is similar to ADB. ACB and DCB are the same angle, so ABC is similar to BDC.

    Therefore, all three triangles are similar.

    Challenge Problem

    G.SRT.5 9. By similar triangles (AA Theorem), between FBC and DBA and between ECB and DCA,

    OD

    OE

    OF

    = =

    = =

    = =

    58 411 2

    3 75

    7 62512

    3 175

    6 6912

    4 95

    ..

    .

    . .

    . .

    By subtraction,

    AE

    AF

    ==

    7 615384615

    6 6

    .

    .

    units

    units

    By similar triangles (AA Theorem), between EBC and DBO and between FCB and DCO,

    OB

    OC

    OD

    = =

    = =

    = =

    914

    12 9239 75

    514

    11 26 25

    58 411 2

    3

    .

    .

    .

    .

    ..

    units

    units

    ..75 units

    By subtraction,

    OA

    OE

    OF

    ===

    8 25

    3 173

    4 95

    .

    .

    .

    units

    units

    units Note: Once you find two side of a right triangle, you can use the Pythagorean Theorem to find the third side. Therefore, there are multiple ways of finding the side lengths.

    LESSON 15: PYTHAGOREAN THEOREM

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    ANSWERS

    ANSWERS

    G.SRT.1 2. Scale Factor Example k

    Center of Dilation

    k > 1

    0 50

    5

    10y

    x

    A

    B

    C'

    A'

    C

    B'

    1.5 (0, 0)

    0 < k < 1

    0 5 15100

    5

    15

    10

    y

    x

    A

    B

    C

    D

    A'B'

    C'D'

    0.25 (0, 10)

    k < 1

    10 5 5

    10

    5

    5y

    x00

    E

    F

    G

    H

    E'

    F'

    G'

    H'

    3 (0, 0)

    LESSON 16: FLOODLIGHTS AND SHADOWS

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    ANSWERS

    G.SRT.2 G.SRT.4

    3.

    LESSON 16: FLOODLIGHTS AND SHADOWS

    Property or Theorem (Lesson Number)

    Definition Explain the Property or Theorem Using Your Own Words or Diagrams

    Parallels Under Dilation Property (Lesson 11)

    A dilation takes a line L to a line L', and line L' is parallel to line L (for any center of dilation not on L).

    The slope of a dilated line is the same for L and L'. For example, when L = 0.5x + 2, then L' = 0.5x + 6.

    5 5 100

    5

    10

    y

    x

    Segment Lengths Under Dilations Property (Lesson 11)

    A dilation with scale factor k takes a segment with length L to a segment with length |k|L.

    A dilated figure will grow or shrink by a factor of k. This figure has a scale factor of 3.

    12

    14

    10

    8

    6

    4

    2

    0

    y

    A

    A'

    C

    C'B

    B'

    x20 10 184 126 148 16

    3

    12

    Angle Measures Under Dilations Property (Lesson 11)

    Any dilation takes an angle to an angle of the same measure.

    When a figure is dilated, the size changes but the angles do not change.

    Correspondence Definition of Similarity (Lesson 12)

    Two polygons are similar if all corresponding angles are congruent and all corresponding side lengths are proportional.

    B

    C

    D

    ES

    P

    Q

    R

    BCPQ

    CDQR

    DERS

    EBSP

    = = =

    (continues)

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    ANSWERSLESSON 16: FLOODLIGHTS AND SHADOWS

    G.SRT.2 G.SRT.4

    3. (continued)

    Property or Theorem (Lesson Number)

    DefinitionExplain the Property or Theorem Using Your Own Words or Diagrams

    Dilation Definition of Similarity (Lesson 12)

    Two figures are similar if one can be transformed into the other by a dilation and, if necessary, a sequence of rigid motions.

    A

    E'

    D'C'B'

    B

    C

    D

    E

    AA Triangle Similarity Theorem (Lesson 13)

    If two triangles have two pairs of congruent angles, the triangles must be similar.

    AA stands for angle-angle. The AA Theorem states that if two angles of one triangle are congruent to the two corresponding angles of another triangle, the triangles are similar. Congruent angles are angles that have the same measure.

    SSS Triangle Similarity Theorem (Lesson 13)

    If two triangles have three pairs of proportional side lengths, the triangles must be similar.

    SSS stands for side-side-side, and proportionality means that all the sides are scaled by the same factor.

    Triangle Proportionality Theorem (Lesson 14)

    A line parallel to one side of a triangle divides the other two sides proportionally.

    43

    E

    15

    129

    20

    D

    A C

    B

    This example shows how the smaller similar triangle is 3

    4 the size of the

    larger triangle.