EM2_G8_M3_Learn_23A_971338_Updated 06.21 by General

A Story of Ratios®

Ratios and Linearity LEARN ▸ Module 3 ▸ Dilations and Similar Figures

Student

Talking Tool Share Your Thinking

I know . . . . I did it this way because . . . . The answer is

because . . . .

My drawing shows . . . . Agree or Disagree

I agree because . . . . That is true because . . . . I disagree because . . . . That is not true because . . . . Do you agree or disagree with

Ask for Reasoning

Why did you . . . ? Can you explain . . . ? What can we do first? How is

Say It Again

related to

I heard you say . . . . said . . . . Another way to say that is . . . . What does that mean?

Content Terms

Place a sticky note here and add content terms.

? Why?

What does this painting have to do with math? Abstract expressionist Al Held was an American painter best known for his “hard edge” geometric paintings. His bright palettes and bold forms create a three-dimensional space that appears to have infinite depth. Held, who sometimes found inspiration in architecture, would often play with the viewer’s sense of visual perception. While most of Held’s artworks are paintings, he also worked in mosaic and stained glass. On the cover Pan North IV, 1985 Al Held, American, 1928–2005 Acrylic on canvas Private collection Al Held (1928–2005), Pan North IV, 1985, acrylic on canvas, 72 x 84 in., private collection. © 2020 Al Held Foundation, Inc./Licensed by Artists Rights Society (ARS), New York

EM2_NA_E_G08_SE_coverpage.indd 1

10/8/20 7:25 AM

Great Minds® is the creator of Eureka Math®, Wit & Wisdom®, Alexandria Plan™, and PhD Science®. Published by Great Minds PBC. greatminds.org Copyright © 2021 Great Minds PBC. All rights reserved. No part of this work may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying or information storage and retrieval systems—without written permission from the copyright holder. Printed in the USA 1 2 3 4 5 6 7 8 9 10 XXX 25 24 23 22 21 ISBN 978-1-64497-133-8

EM2_0803SE_title_page.indd 2

3/4/2021 5:32:00 PM

A Story of Ratios®

Ratios and Linearity ▸ 8 LEARN

Module

EM2_0803SE_title_page.indd 1

Scientific Notation, Exponents, and Irrational Numbers

Rigid Motions and Congruent Figures

Dilations and Similar Figures

Linear Equations in One and Two Variables

Systems of Linear Equations

Functions and Bivariate Statistics

9/15/2020 6:21:41 PM

EUREKA MATH2

8 ▸ M3

Contents Dilations and Similar Figures Topic A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Lesson 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Dilations

Exploring Angles in Similar Triangles

Lesson 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Lesson 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Exploring Dilations

Similar Triangles

Lesson 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Enlargements

Topic D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Lesson 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Properties of Dilations Lesson 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Using Lined Paper to Explore Dilations Lesson 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Figures and Dilations Lesson 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 The Shadowy Hand Lesson 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Dilations on a Grid Lesson 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Dilations on the Coordinate Plane

Topic C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121

Lesson 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Describing Dilations Lesson 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Sequencing Transformations Lesson 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

EM2_0803SE_content_page.indd 2

Using Similar Figures to Find Unknown Side Lengths

Lesson 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Applications of Similar Figures Lesson 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Similar Right Triangles Lesson 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Similar Triangles on a Line

Resources Mixed Practice 1 . . . . . . . . . . . . . . . . . . . . . . . . . 279 Mixed Practice 2 . . . . . . . . . . . . . . . . . . . . . . . . . 283 Fluency Resources

Similar Figures

Applications of Similar Figures Lesson 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Reductions and More Enlargements

Topic B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215

Lesson 10 (Transformation) . . . . . . . . . . . . . . . . . Sprint: Apply Properties of Exponents for a Power Raised to a Power . . . . . . . . . . . . . . Sprint: Compare Numbers and Square Roots. . . Sprint: One-Step Equations—Multiplication. . . Sprint: Solve Simple Proportions . . . . . . . . . . . . Sprint: Solve Two-Step Equations . . . . . . . . . . .

289 291 295 299 303 307

Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 312

10/10/2020 6:33:16 PM

Dilations

TOPIC

Alice in Wonderland Reads the Nutrition Label

In the classic novel Alice’s Adventures in Wonderland—written, it is worth noting, by a mathematician— the character of Alice finds herself changing in size. A lot.

EM2_0803SE_A_topic_opener.indd 3

10/30/2020 5:22:31 AM

8 ▸ M3 ▸ TA

EUREKA MATH2

First, she shrinks. Then, she grows. Then, she shrinks again. Then, she grows again. And so it goes, each shrinking or growing caused by a different food or drink she has consumed. It all fits the spirit of the Alice books: zany, impossible, and illogical. Or is shrinking and growing really that impossible? Shrinking and growing aren’t such ridiculous processes. They can be described by using the simple and powerful mathematics of dilations. If Alice had just read the fine print a little more carefully, perhaps her strange adventures would not have seemed so strange!

TO P I C O P E N E R

EM2_0803SE_A_topic_opener.indd 4

10/9/2020 11:38:57 AM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 1

LESSON Name

Date

Exploring Dilations Dilation—A Transformation

Exploring Dilations

Comparing Dilations and Scale Drawings

EM2_0803SE_A_L01_classwork.indd 5

9/15/2020 12:14:39 PM

EM2_0803SE_A_L01_classwork.indd 6

9/15/2020 12:14:39 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 1

EXIT TICKET Name

Date

For problems 1 and 2, determine whether the transformation is a rigid motion or is not a rigid motion. Explain how you know. 1.

Aʹ

Bʹ O

EM2_0803SE_A_L01_exit_ticket.indd 7

9/15/2020 12:16:24 PM

EM2_0803SE_A_L01_exit_ticket.indd 8

9/15/2020 12:16:24 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 1

RECAP Name

Date

Exploring Dilations In this lesson, we •

classified a dilation as a transformation that is not a rigid motion.

•

described the effects of changing the center and scale factor of a dilation.

Terminology A transformation is a rule that maps each point of the plane to a unique point in the plane.

Examples For problems 1–4, name the transformation. Then state whether the transformation is a rigid motion or is not a rigid motion. 2.

Translation The transformation is a rigid motion. If the distance between any two points changes, then the transformation is not a rigid motion.

Reflection The transformation is a rigid motion. If the distance between any two points stays the same, then the transformation is a rigid motion.

EM2_0803SE_A_L01_lesson_recap.indd 9

Rotation The transformation is a rigid motion. 9

10/10/2020 1:47:10 PM

EUREKA MATH2

8 ▸ M3 ▸ TA ▸ Lesson 1

5. Each diagram shows a dilation with center O and scale factor  3 , 1, or 2. Write the scale factor by the corresponding diagram. If the image is larger than the figure, then the scale factor is greater than 1.

If the image is the same size as the figure, then the scale factor is equal to 1.

O Cʹ

C Cʹ

Bʹ

If the image is smaller than the figure, then the scale factor is less than 1.

A Aʹ

Bʹ

Aʹ C

B Bʹ

_1   3

Aʹ

R ecap

EM2_0803SE_A_L01_lesson_recap.indd 10

10/10/2020 1:47:10 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 1

PRACTICE Name

Date

1. Is a dilation a rigid motion? Explain.

2. Classify the following transformations by writing them in the correct location on the diagram. •

Reflections

•

Dilations

•

Rotations

•

Translations

Transformations Rigid Motions

3. Circle the transformations that are also rigid motions.

4. Name each transformation shown. Then circle the transformations that are also rigid motions.

EM2_0803SE_A_L01_daily_practice.indd 11

10/9/2020 11:50:35 AM

EUREKA MATH2

8 ▸ M3 ▸ TA ▸ Lesson 1

For problems 5–8, name the transformation shown in the diagram.

Hʹ

Gʹ

Kʹ Hʹ

Kʹ

Gʹ

H G

Jʹ

J J

Hʹ

Gʹ

Jʹ

Kʹ Kʹ H G

Hʹ

Jʹ

K J

Gʹ

P ractice

EM2_0803SE_A_L01_daily_practice.indd 12

10/9/2020 11:50:35 AM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 1

9. Four images of △T are shown. State which triangle does not represent the image of △T under a dilation.

A B T C

Remember For problems 10–13, evaluate the expression for the given value. 10. 4x + 5 when x = 3

11. 8x − 1 when x = 5

12. 9x + 6 when x = −2

13. 5x − 8 when x = −4

EM2_0803SE_A_L01_daily_practice.indd 13

P ractice

10/9/2020 11:50:36 AM

EUREKA MATH2

8 ▸ M3 ▸ TA ▸ Lesson 1

14. Consider the diagram.

58° 72° (14x – 38)°

What is the value of x ?

15. Is figure ABCD congruent to figure EFGH ? If so, describe a sequence of rigid motions that maps figure ABCD onto figure EFGH. If not, explain how you know. y 8 7 6

5 4

3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

B 1

−2 −3 −4 −5

G H

−6 −7 −8

P ractice

EM2_0803SE_A_L01_daily_practice.indd 14

10/9/2020 11:50:36 AM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 2

LESSON Name

Date

Enlargements 1. The diagram shows figures A, B, and C and point O. Figure B is the image of figure A under a dilation with center O and scale factor 2. Figure C is the image of figure A under a dilation with center O and scale factor 4. Sketch the image of figure A under a dilation with center O and scale factor 3. Label this figure D.

B A O

Enlarge 2. Draw and label the image of point P under a dilation with center O and scale factor 4.

EM2_0803SE_A_L02_classwork.indd 15

10/10/2020 1:49:25 PM

EUREKA MATH2

8 ▸ M3 ▸ TA ▸ Lesson 2

3. Draw and label the image of point Q under a dilation with center O and scale factor 2.

4. Draw and label the image of point P under a dilation with center O and scale factor 3.

P O

5. Describe the dilation shown in the diagram that maps point R to R′.

O R

× Rʹ

L esson

EM2_0803SE_A_L02_classwork.indd 16

10/10/2020 1:49:26 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 2

6. Describe the dilation shown in the diagram that maps point Q to Q′.

Q × × × ×

Qʹ

Distance Relationships Under Dilations 7. Write an equation that represents the dilation shown with center O and scale factor r that maps point P to P′.

O P

Pʹ

EM2_0803SE_A_L02_classwork.indd 17

Lesson

11/18/2020 11:24:06 AM

EUREKA MATH2

8 ▸ M3 ▸ TA ▸ Lesson 2

8. A dilation with center O and scale factor 6 maps point R to point R′. What is the distance between O and R′?

2 ft

Rʹ

9. Point P′ is the image of point P under a dilation with center O. What is the scale factor of the dilation?

Pʹ 6.4 cm P O

L esson

EM2_0803SE_A_L02_classwork.indd 18

3.2 cm

10/10/2020 1:49:27 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 2

Features of Dilations ¯ under a dilation with center O. 10. Yu Yan draws  ¯ S′H′ . She claims it is the image of   SH  Do you agree with Yu Yan? Explain.

Sʹ

Hʹ

S H O

EM2_0803SE_A_L02_classwork.indd 19

L esson

10/10/2020 1:49:27 PM

EUREKA MATH2

8 ▸ M3 ▸ TA ▸ Lesson 2

11. Sara draws △B′I′T′. She claims it is the image of △BIT under a dilation with center O. Do you agree with Sara? Explain.

Bʹ B O I

Iʹ Tʹ

Dilations

Pʹ P O A dilation with center

and scale factor center name

Example: A dilation with center

L esson

EM2_0803SE_A_L02_classwork.indd 20

maps a figure to its image. value

and scale factor

maps P to P′.

10/10/2020 1:49:27 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 2

EXIT TICKET Name

Date

Points O and P are shown.

a. Draw the image of point P under a dilation with center O and scale factor 3. Label the image P′. b. If the distance between O and P is 5 ft, what is the distance between O and P′?

EM2_0803SE_A_L02_exit_ticket.indd 21

10/9/2020 12:11:36 PM

EM2_0803SE_A_L02_exit_ticket.indd 22

10/9/2020 12:11:36 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 2

RECAP Name

Date

Enlargements In this lesson, we •

used a transparency to apply a dilation with a whole-number scale factor greater than 1.

•

described the effects of a dilation with a whole-number scale factor greater than 1.

Examples 1. A dilation with center O and scale factor 4 maps point K to point K′.

Terminology A dilation is a transformation with a center, called the center of dilation, and a scale factor greater than 0 that maps a figure to its image. A dilation with center O and scale factor r, with r > 0, maps any point P that is not O to a point P′ with the following features:

⟶

• P′ is on  OP 

• OP′ = r · OP

0.8 ft 0.8 ft × Kʹ 0.8 ft × 0.8 ft

• The center of dilation O and its image, O′, are the same point.

⟶

First, draw OK    which starts at center O and extends beyond K.

Then, use a transparency to mark the distance between O and K. Mark this ⟶ distance 3 more times on OK    to locate K′.

a. Draw and label K′. b. What is the distance between O and K′?

OK′ = r · OK OK′ = 4 · 0.8

The distance between O and K′ is 4 times the distance between O and K.

OK′ = 3.2 The distance between O and K′ is 3.2 ft.

EM2_0803SE_A_L02_lesson_recap.indd 23

04-Sep-21 11:16:44 AM

EUREKA MATH2

8 ▸ M3 ▸ TA ▸ Lesson 2

2. A dilation with center O maps point Y to point Y′. What is the scale factor of the dilation?

2 in Yʹ Y 2 3

The distance between O and Y′ is 3 times the distance between O and Y.

OY′ = r · OY 2 = r · 2_3 3=r

The scale factor of the dilation is 3.

R ecap

EM2_0803SE_A_L02_lesson_recap.indd 24

10/10/2020 1:50:57 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 2

PRACTICE Name

Date

1. Circle an answer choice in each box to make a true statement.

M I T A dilation centered at

(A)

with scale factor A point T point M

EM2_0803SE_A_L02_daily_practice.indd 25

(B)

maps point I to point E. B

3 4 5

11/5/2020 11:59:53 AM

EUREKA MATH2

8 ▸ M3 ▸ TA ▸ Lesson 2

2. Sara provides the following description of the dilation shown.

S U

Uʹ A dilation with center S and scale factor 2 maps point U to point U′. I know this because the distance between U and U′ is two times the distance between U and S. Do you agree with Sara? Explain.

P ractice

EM2_0803SE_A_L02_daily_practice.indd 26

10/9/2020 1:02:08 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 2

For problems 3–5, draw and label the image of point K under a dilation with center O and the given scale factor. 3. Scale factor 2

K O

4. Scale factor 4

5. Scale factor 7

O K

EM2_0803SE_A_L02_daily_practice.indd 27

P ractice

10/9/2020 1:02:08 PM

EUREKA MATH2

8 ▸ M3 ▸ TA ▸ Lesson 2

6. A dilation with center O and scale factor 5 maps point G to point G′.

4 in O G

Gʹ a. What is the distance from point O to point G′?

b. What is the distance from point G to point G′?

7. A dilation with center O maps point B to point B′. What is the scale factor of the dilation? Explain.

O 1.1 ft B

5.5 ft

Bʹ

P ractice

EM2_0803SE_A_L02_daily_practice.indd 28

10/9/2020 1:02:08 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 2

8. A dilation with center O maps point H to point H′. The distance between O and H′ is 24 cm. The distance between O and H is 3 cm. What is the scale factor of the dilation?

Remember For problems 9–12, evaluate the expression for the given value. 9. 3x + 5 when x =  _ 

10. 18x − 2 when x =  _ 

1 3

1 2

11. 6x + 2 when x = −  _ 

12. 15x − 4 when x = −  _ 

1 3

1 5

13. Describe a sequence of rigid motions that shows △ ABC ≅ △JKL.

EM2_0803SE_A_L02_daily_practice.indd 29

P ractice

10/9/2020 1:02:09 PM

EM2_0803SE_A_L02_daily_practice.indd 30

10/9/2020 1:02:09 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 3

LESSON Name

Date

Reductions and More Enlargements 1. Sketch the image of figure A under a dilation with center O and scale factor _  2  . 3

Reduce 2. Draw and label the image of point P under a dilation with center O and scale factor _  2  . 3

EM2_0803SE_A_L03_classwork.indd 31

10/9/2020 1:06:24 PM

EUREKA MATH2

8 ▸ M3 ▸ TA ▸ Lesson 3

3. Draw and label the image of point P under a dilation with center O and scale factor _  1  . 2

P O

4. Draw and label the image of point R under a dilation with center O and scale factor _  3  . 4

5. In the diagram, Eve identified the scale factor as −2 for the dilation centered at point O. Jonas identified the scale factor as 2 for the dilation. Do you agree with Eve or Jonas? Why?

Q 9.5 cm Qʹ

4.75 cm O

LESSON

EM2_0803SE_A_L03_classwork.indd 32

10/9/2020 1:06:24 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 3

Reduce or Enlarge? 6. A dilation with center O and scale factor _  2  maps point R to point R′. What is the distance 3 between O and R′? Write and solve an equation to support your answer.

R Rʹ

6 ft O

7. A dilation with center O maps point P to point P′. What is the scale factor of the dilation? Write and solve an equation to support your answer.

Pʹ 6.825 cm P O

EM2_0803SE_A_L03_classwork.indd 33

2.1 cm

L esson

10/9/2020 1:06:25 PM

EUREKA MATH2

8 ▸ M3 ▸ TA ▸ Lesson 3

8. Draw and label the image of point Q under a dilation with center O and scale factor 2.5.

Scale Factor Match

Diagram Card Letter

Scale Factor Card Number

L esson

EM2_0803SE_A_L03_classwork.indd 34

10/9/2020 1:06:25 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 3

EXIT TICKET Name

Date

Points O and P are shown.

a. Draw the image of point P under a dilation with center O and scale factor _  1 . Label 2 the image P′. b. If the distance between O and P is 9 ft, what is the distance between O and P′ ? Write and solve an equation to support your answer.

EM2_0803SE_A_L03_exit_ticket.indd 35

10/9/2020 1:10:21 PM

EM2_0803SE_A_L03_exit_ticket.indd 36

10/9/2020 1:10:21 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 3

RECAP Name

Date

Reductions and More Enlargements In this lesson, we •

used a ruler to apply a dilation with a scale factor greater than 0.

•

described the effects of a dilation with any scale factor greater than 0.

Examples

1. Draw and label the image of point R under a dilation with center O and scale factor _  1 . 4

Rʹ

⟶ First, use a ruler to draw  OR 

R Then use the equation to find the distance between O and R′.

OR′ = r ⋅ OR

and measure the distance from O to R, which is 6 cm.

OR′ = _1   ⋅ 6

To locate R′, use a ruler to measure 1.5 cm from ⟶ O on OR    and draw R′.

OR′ = 1.5

2. A dilation with center O and scale factor _  2 maps point S to point S′. What is the distance 3 between O and S′ ? Write and solve an equation to support your answer. 3 4

Sʹ S

OS′ = r ⋅ OS

OS′ = _2 ⋅ _3

A scale factor less than 1 results in an image closer to the center of dilation.

3 4

The distance between O and S′ is  _1 yd.

OS′ = _1 2

EM2_0803SE_A_L03_lesson_recap.indd 37

10/9/2020 1:12:03 PM

EUREKA MATH2

8 ▸ M3 ▸ TA ▸ Lesson 3

3. A dilation with center O maps point T to point T′. What is the scale factor of the dilation? Write and solve an equation to support your answer.

Tʹ 2 ft

T OT′ = r ⋅ OT

0.625 ft

2 = r ⋅ 0.625

____   2 = r 0.625

3.2 = r The scale factor of the dilation is 3.2.

R ecap

EM2_0803SE_A_L03_lesson_recap.indd 38

The scale factor should be greater than 1 because T′ is farther from center O than T is from center O.

10/9/2020 1:12:03 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 3

PRACTICE Name

Date

1. Each diagram in the table shows a dilation with center O. Indicate whether the scale factor is greater than 1, between 0 and 1, or equal to 1. Greater than 1

Diagram

Between 0 and 1

Equal to 1

Pʹ

P O P Pʹ O P

Pʹ

O P

Pʹ

EM2_0803SE_A_L03_daily_practice.indd 39

10/10/2020 1:52:27 PM

EUREKA MATH2

8 ▸ M3 ▸ TA ▸ Lesson 3

2. Draw and label the image of point P under a dilation with center O and scale factor  _1  . 3

3. Draw and label the image of point Q under a dilation with center O and scale factor  _1  . 5

P ractice

EM2_0803SE_A_L03_daily_practice.indd 40

10/10/2020 1:52:27 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 3

4. Draw and label the image of point R under a dilation with center O and scale factor  _3  . 4

For problems 5–7, use the equation OR′ = r ⋅ OR to calculate the answer.

5. A dilation with center O and scale factor  _1  maps point R to point R′. What is the distance 3 between O and R′ ?

O Rʹ 4.5 in

EM2_0803SE_A_L03_daily_practice.indd 41

P ractice

10/10/2020 1:52:28 PM

EUREKA MATH2

8 ▸ M3 ▸ TA ▸ Lesson 3

6. A dilation with center O and scale factor  _2  maps point R to point R′. What is the distance 5 between O and R?

R O

Rʹ 5 yards

7. A dilation with center O maps point R to point R′. What is the scale factor of the dilation?

O 18 in

24 in

Rʹ R

P ractice

EM2_0803SE_A_L03_daily_practice.indd 42

10/10/2020 1:52:28 PM

EUREKA MATH2 8 ▸ M3 ▸ TA ▸ Lesson 3

Remember 8. 5x + 2 when x = 0.3

9. 4x − 5 when x = 0.7

10. 9x + 1 when x = −0.2

11. 3x − 5 when x = −1.4

For problems 8–11, evaluate the expression for the given value.

12. Find the length of  AB  ‾. y

10 9 8 7 6 5 4 3 2 1

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10

EM2_0803SE_A_L03_daily_practice.indd 43

1 2 3 4 5 6 7 8 9 10

P ractice

10/10/2020 1:52:28 PM

EUREKA MATH2

8 ▸ M3 ▸ TA ▸ Lesson 3

13. The coordinate plane shows the locations of Nora’s and Vic’s houses. Nora travels the path shown to get to Vic’s house. House Locations 8

Distance (kilometers)

6 5

Vic’s House

3 2 1

Nora’s House

Distance (kilometers)

a. Write the locations of Nora’s and Vic’s houses as ordered pairs. b. How many kilometers is the path that Nora takes to Vic’s house?

P ractice

EM2_0803SE_A_L03_daily_practice.indd 44

10/10/2020 1:52:28 PM

Properties of Dilations

TOPIC

Shadow Play

If you’ve ever made shadow puppets on a distant wall, you know that your shadow can be much bigger than you are. When lit from behind with a flashlight, an ordinary hand that is 6 inches long can look like a 3-foot-long hand of a giant. Why? The size of a shadow has to do with geometry. Where is the light source? How far is it from the person casting the shadow to the surface on which the shadow is cast? Understanding the mathematical process of dilation is crucial to figuring out the size of a shadow. And the upcoming topic will—if you can forgive the pun—shed some light on this process.

EM2_0803SE_B_topic_opener.indd 45

10/9/2020 1:22:00 PM

EM2_0803SE_B_topic_opener.indd 46

10/9/2020 1:22:00 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 4

LESSON Name

Date

Using Lined Paper to Explore Dilations ¯ under a dilation with center O. 1. Dylan claims that the diagram shows the image of  AB 

Aʹ A O

Bʹ

¯ and   ¯ a. What do you notice about  AB  AʹBʹ ? b. Do you agree with Dylan? Explain.

Segments on Lined Paper 2. Summarize how to draw the image of a segment under a dilation.

EM2_0803SE_B_L04_classwork.indd 47

10/30/2020 5:23:58 AM

EM2_0803SE_B_L04_classwork.indd 48

10/9/2020 1:23:13 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 4

EXIT TICKET Name

Date

11 . The diagram shows a dilation with center O and scale factor  __ 7

Pʹ

Qʹ

¯. Draw and label the image of R under a dilation with center O and a. Place a point R on   PQ  11 scale factor  __ . 7

b. How do you know the image of R is in the correct location?

EM2_0803SE_B_L04_exit_ticket.indd 49

10/9/2020 1:25:51 PM

EM2_0803SE_B_L04_exit_ticket.indd 50

10/9/2020 1:25:51 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 4

RECAP Name

Date

Using Lined Paper to Explore Dilations In this lesson, we •

drew the image of a segment under a dilation.

•

identified that under a dilation •

lines map to lines,

•

angles map to angles of the same measure,

•

segments map to segments of proportional length,

•

a line and its image are parallel when the center is not on the line, and

•

a segment and its image are parallel when the center is not on the line that contains the segment.

Examples ¯ onto   ¯ 1. A dilation is drawn on lined paper. The dilation has center O and maps  AC  AʹCʹ .

The ray from center O to point A crosses 3 spaces. From center O to point Aʹ, the ray crosses 10 spaces.

Aʹ

Bʹ

Cʹ Draw a ray from center O through point B. Point Bʹ is at the intersection of the ray and AʹCʹ   ¯.

EM2_0803SE_B_L04_lesson_recap.indd 51

10/30/2020 5:46:40 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 4

a. What is the scale factor of the dilation?

OAʹ = r ⋅ OA 10 = r ⋅ 3

__  = r 10 3

10 __

The scale factor is  3  . ¯. b. Place a point B on   AC 

c. Draw and label the image of point B under the same dilation. ¯ onto   ¯ 2. A dilation is drawn on lined paper. The dilation has center O and maps  PQ  PʹQʹ . ∠OPQ and ∠OPʹQʹ are

congruent because they are corresponding angles created by the ⟶ transversal ray  OP  and parallel segments, PQ    ¯ ¯. and  PʹQʹ 

Pʹ

¯ and   ¯   PQ  PʹQʹ  are parallel because they both lie on lines of the lined paper.

Qʹ There are 12 spaces between O and Q, and there are 8 spaces between O and Q ʹ. So the scale factor is

OQʹ __ ___   =   8   , or _ 2  . OQ

¯ and   ¯ a. What is the relationship between  PQ  PʹQʹ ? Between ∠OPQ and ∠OPʹQʹ ?

PQ    ¯ and   ¯ PʹQʹ  are parallel. ∠OPQ and ∠OPʹQʹ are congruent.

¯ and the length b. Write an equation that represents the relationship between the length of  PQ  PʹQʹ . of   ¯ PʹQʹ = 2_   ⋅ PQ 3

R e ca p

EM2_0803SE_B_L04_lesson_recap.indd 52

10/30/2020 5:31:49 AM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 4

PRACTICE Name

Date

1. A dilation is drawn on lined paper. The dilation has center O and maps point R to point Rʹ.

Rʹ

What is the scale factor of the dilation? 3_ A. 7 

EM2_0803SE_B_L04_daily_practice.indd 53

_7 B.   3

10 C. __ 3

3 D.  __ 10

10/31/2020 11:42:59 AM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 4

2. Several dilations are drawn on lined paper. Each dilation has center O and maps point L to

point Lʹ. Find each scale factor and write it by the corresponding diagram. Choose from the

_3 _5 _8

following scale factors:  8  ,  8  ,  5  , and  3  .

L L

Lʹ

Lʹ Lʹ

P ractice

EM2_0803SE_B_L04_daily_practice.indd 54

10/9/2020 2:43:44 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 4

3. Several dilations are drawn on lined paper. Each dilation has center O and maps point R to

point Rʹ. Find each scale factor and write it by the corresponding diagram. Choose from the

_1 _8 _9

following scale factors:  9  ,  9  ,  8  , and 9. a.

Rʹ

R Rʹ

EM2_0803SE_B_L04_daily_practice.indd 55

Rʹ R

P ractice

10/9/2020 2:43:45 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 4

4. Points A, B, and E are drawn on lined paper as shown.

a. Draw and label the images of B and E under a dilation with center A and scale factor  9  . b. Draw a line segment that connects B to E and a line segment that connects Bʹ to Eʹ. ¯ and   ¯ BʹEʹ  parallel? Explain. c. Are   BE 

d. Is the value of ____   BE greater than 1, or is the value between 0 and 1? How do you know? BʹEʹ

¯ and e. Write an equation that represents the relationship between the length of  BE  ¯ the length of   BʹEʹ .

P ractice

EM2_0803SE_B_L04_daily_practice.indd 56

10/9/2020 2:43:45 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 4

¯ onto   ¯ 5. A dilation is drawn on lined paper. The dilation has center T and maps  RS  RʹSʹ .

Rʹ R

Sʹ S

T RʹSʹ ____ greater than 1, or is the value between 0 and 1? Explain.

a. Is the value of

¯ and the length b. Write an equation that represents the relationship between the length of  RS  ¯ of   RʹSʹ .

¯ onto   ¯ 6. A dilation is drawn on lined paper. The dilation has center O and maps  PQ  PʹQʹ .

P Pʹ

Q Qʹ

a. What is the scale factor of the dilation?

EM2_0803SE_B_L04_daily_practice.indd 57

P ractice

10/9/2020 2:43:46 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 4

8.25 ¯ as 6.25 cm and the length of  ¯ b. Sara measures the length of  PQ  PʹQʹ  as 8.25 cm. Is ___  6.25 

equivalent to the scale factor you found in part (a)? If not, why do you think they are different?

¯ onto   ¯ PʹTʹ . 7. A dilation is drawn on lined paper. The dilation has center C and maps  PT 

C Pʹ

Tʹ

¯ and   ¯ a. Are   PT  PʹTʹ  parallel? How do you know?

¯ and the length b. Write an equation that represents the relationship between the length of  PT  ¯ of   PʹTʹ . c. The measure of ∠TPC is 27°. What is the measure of ∠TʹPʹC ? Explain. d. Nora explains that ∠CTʹPʹ is not congruent to ∠CTP because there is no rigid motion that maps ∠CTʹPʹ onto ∠CTP. Do you agree with Nora? Why?

P ractice

EM2_0803SE_B_L04_daily_practice.indd 58

10/9/2020 2:43:46 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 4

8. The diagram shows a dilation with center O and scale factor  2  .

Sʹ

R Rʹ

¯? ¯ is 9 cm, what is the length of  OSʹ  a. If the length of  OS 

¯ is 1.5 cm, what is the length of  OR  ¯? b. If the length of  ORʹ 

¯ and the length c. Write an equation that represents the relationship between the length of  RS  ¯ of   RʹSʹ .

9. The diagram shows a dilation with center O and scale factor 1.8.

Fʹ

F I

Xʹ

a. Draw and label the image of point I under the same dilation.

EM2_0803SE_B_L04_daily_practice.indd 59

P ractice

10/9/2020 2:43:46 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 4

¯? ¯ is 18 mm, what is the length of  FʹIʹ  b. If the length of  FI 

¯ is 9 mm, what is the length of  IX  ¯? c. If the length of  IʹXʹ 

10. The diagram shows a dilation with center O and scale factor  4  .

Pʹ

T Bʹ

a. Draw and label the image of point T under the same dilation.

¯ is   inches, what is the length of  ¯ BʹTʹ ? b. If the length of  BT  4

¯? c. If the length of  ¯ PʹTʹ  is  4 inches, what is the length of  PT 

d. What is the value of ___  BP ? How do you know? BʹPʹ

P ractice

EM2_0803SE_B_L04_daily_practice.indd 60

10/9/2020 2:43:47 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 4

Remember For problems 11–14, evaluate the expression for the given value. 3   when x = 4 11. 8x + _ 4

1 12. 5x − _   when x = 7 2

1 13. 6x + _   when x = −3 4

5 14. 2x − _   when x = −2 6

15. Is the transformation shown in the diagram a rigid motion? Explain.

Cʹ O

EM2_0803SE_B_L04_daily_practice.indd 61

P ractice

10/9/2020 2:43:47 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 4

16. A segment has endpoints (−2, 4) and (4, 4). a. Plot the points and create the segment in the coordinate plane. y 8 7 6 5 4 3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

−2 −3 −4 −5 −6 −7 −8

b. What is the length of the segment?

P ractice

EM2_0803SE_B_L04_daily_practice.indd 62

10/9/2020 2:43:47 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 5

LESSON Name

Date

Figures and Dilations 1. Draw and label the image of △ULB under a dilation with center O and scale factor 2.

EM2_0803SE_B_L05_Classwork.indd 63

10/9/2020 2:48:56 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 5

Polygon Enlargements and Reductions 2. Draw and label the image of figure JEANS under a dilation with center O and scale factor 3.

N A

E J

L esson

EM2_0803SE_B_L05_Classwork.indd 64

10/9/2020 2:48:57 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 5

3. Consider the figure VEST.

a. Draw and label the image of figure VEST under a dilation with center V and scale factor  4 .

V E T

S b. Measure and record the segment lengths in the table. Figure VEST

Figure VʹEʹSʹTʹ

VE =

VʹEʹ =

ES =

EʹSʹ =

ST =

SʹTʹ =

TV =

TʹVʹ =

c. Select measurements and show the calculation that demonstrates you have applied the scale factor of the dilation correctly.

EM2_0803SE_B_L05_Classwork.indd 65

L esson

10/9/2020 2:48:57 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 5

The Business of Dilations 4. In the diagram, a dilation with center O and scale factor 2 maps figure RS onto figure RʹSʹ.

Rʹ

Sʹ

Draw and label the image of figure RS under a dilation with center O and scale factor 3. Use the first diagram as a model to help you.

Lesson

EM2_0803SE_B_L05_Classwork.indd 66

04-Sep-21 10:29:27 AM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 5

The Business of Dilations Activity An ice cream store owner wants to create a reduction of the store’s logo to make business cards and an enlargement of the logo to make flyers. •

Determine which partner will create the flyer image and which partner will create the business card image.

•

Choose tools for your dilation. For the business card, get the Logo Business Card page and any tools you need to create the image. Your image must be able to fit inside the business card shown on the page. Actual size of business card

For the flyer, get a sheet of construction paper, the Logo Flyer quarter-page, tape, and any tools you need to create the image. Tape the quarter-page to the top edge of the construction paper. Your image must fit on the construction paper. O

•

With your partner, determine the scale factor to use for the reduction from center O and the scale factor to use for the enlargement from center O.

•

Draw the image of the logo.

Calculations:

EM2_0803SE_B_L05_Classwork.indd 67

Lesson

07-Sep-21 12:49:20 PM

EM2_0803SE_B_L05_Classwork.indd 68

10/9/2020 2:48:58 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 5

EXIT TICKET Name

Date

1. Figure AʹB′C′D′ is the image of figure ABCD under a dilation with center O and scale factor 3.

Bʹ

Cʹ Dʹ B D

C Aʹ

What is the value of ____   B′C′? Explain how you know. BC

2. Draw the image of figure AB under a dilation with center O and scale factor 2.

EM2_0803SE_B_L05_exit_ticket.indd 69

10/9/2020 2:50:04 PM

EM2_0803SE_B_L05_exit_ticket.indd 70

10/9/2020 2:50:04 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 5

RECAP Name

Date

Figures and Dilations In this lesson, we •

drew images of polygons under dilations.

•

drew images of figures with curves under dilations.

Examples

1. Draw and label the image of △ ABC under a dilation with center O and scale factor 3.

Aʹ B Find the image of each vertex of the triangle. Then connect the images of the vertices to draw the image of △ ABC.

Cʹ

Bʹ

2. Draw and label the image of figure PQRSTU under a dilation with center P and scale factor _  1  . 3

Q The length of the image of a segment is   1  the 3 length of the segment.

Rʹ Lʹ Sʹ

Qʹ

EM2_0803SE_B_L05_lesson_recap.indd 71

Mʹ P

Tʹ

Uʹ

Because line segments cannot be used to connect points R, S, and T, the images of many points along the curve must be found for an accurate dilation.

U 71

10/30/2020 5:33:09 AM

EM2_0803SE_B_L05_lesson_recap.indd 72

10/9/2020 3:34:05 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 5

PRACTICE Name

Date

1. A dilation with center O maps △ XYZ onto △ XʹYʹZʹ.

Yʹ X

4.5 Xʹ

Zʹ

7.5 O

a. For each side in △ XYZ, find the value of the ratio of the length of the side’s image to the length of the side.

b. What is the scale factor of the dilation? Explain.

EM2_0803SE_B_L05_daily_practice.indd 73

10/10/2020 2:44:38 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 5

2. Draw and label the image of figure LMNO under a dilation with center P and scale factor 2.

L M

O N

3. Draw and label the image of figure HEARTS under a dilation with center H and scale factor 2.

T S

P ractice

EM2_0803SE_B_L05_daily_practice.indd 74

10/10/2020 2:44:38 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 5

4. Ava is creating business cards with a T-shirt logo. She applies a dilation with center A to reduce the size of the logo. What is the scale factor of the dilation?

EM2_0803SE_B_L05_daily_practice.indd 75

P ractice

10/10/2020 2:44:39 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 5

5. Kabir’s Cookie Shop is creating flyers with a cookie-shaped logo. Kabir applies a dilation with center C and scale factor 3 to enlarge the size of the logo. He finds the image of five points. What is a problem Kabir could have with this dilation?

Remember For problems 6–9, evaluate the expression for the given value. 6. 6x + _3   when x = _1   4 4

7. 4x − _3   when x = _1   5 5

8. 7x + _2   when x = −_1  3 3

9. 8x − _1   when x = −_1  4 4

P ractice

EM2_0803SE_B_L05_daily_practice.indd 76

10/10/2020 2:44:39 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 5

10. Draw and label the image of point P under a dilation with center O and scale factor 2.

P O

11. A segment has endpoints (2, 0) and (2, 6). a. Plot the points and create the segment in the coordinate plane. y 8 7 6 5 4 3 2 1 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

−2 −3 −4 −5 −6 −7 −8

b. What is the length of the segment?

EM2_0803SE_B_L05_daily_practice.indd 77

P ractice

10/10/2020 2:44:39 PM

EM2_0803SE_B_L05_daily_practice.indd 78

10/10/2020 2:44:39 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 6

LESSON Name

Date

The Shadowy Hand 1. Focus Question:

Describe the outcome you are trying to achieve. Include a sketch.

Where to Stand 2. Predict how far your hand should be from the light source.

3. Record any measurements your group makes.

EM2_0803SE_B_L06_classwork.indd 79

10/10/2020 2:46:58 PM

8 ▸ M3 ▸ TB ▸ Lesson 6

EUREKA MATH2

4. Calculate how far your group member’s hand should be from the light source.

5. Test your predicted and calculated distances by standing in these locations and recording the results. Include a sketch of your results for your predicted and calculated distances.

Lesson

EM2_0803SE_B_L06_classwork.indd 80

10/10/2020 3:59:47 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 6

6. Measure and record the actual distance your group member’s hand should be from the light source. 7. Describe your results. Compare your results from the predicted, calculated, and actual distances.

In a Different Light (Optional) 8. Predict how far your group member’s hand should be from the light source.

9. Record any new measurements your group has made.

10. Calculate how far your group member’s hand should be from the light source.

EM2_0803SE_B_L06_classwork.indd 81

L esson

10/10/2020 2:46:58 PM

8 ▸ M3 ▸ TB ▸ Lesson 6

EUREKA MATH2

11. Test your predicted and calculated distances by standing in these locations and recording the results. Include a sketch of your results for your predicted and calculated distances.

12. Measure and record the actual distance your group member’s hand should be from the light source.

13. Compare your results from the predicted, calculated, and actual distances. Was your predicted distance closer to the actual distance this time?

L esson

EM2_0803SE_B_L06_classwork.indd 82

10/10/2020 2:46:59 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 6

EXIT TICKET Name

Date

Reflect on the lesson.

EM2_0803SE_B_L06_exit_ticket.indd 83

10/9/2020 3:59:00 PM

EM2_0803SE_B_L06_exit_ticket.indd 84

10/9/2020 3:59:00 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 6

PRACTICE Name

Date

1. An activity is set up as shown in the diagram. The figure is not drawn to scale. •

A light source is placed 30 feet from a wall and turned on.

•

On the wall is an outline of a hand. The height of the hand from the wrist to the tip of the longest finger is 15 inches.

•

When your classmate’s hand is held parallel to the wall, the height of their hand from the wrist to the tip of their longest finger is 6 inches.

a. Label the diagram with the given measurements. b. How far should your classmate’s hand be from the light source so their hand’s shadow matches the hand outline on the wall?

c. Explain your answer by using dilations.

EM2_0803SE_B_L06_daily_practice.indd 85

10/10/2020 7:08:53 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 6

Remember For problems 2–5, evaluate the expression for the given value. 2. 0.4x + 5 when x = 8

3. 1.2x − 4 when x = 4

4. 0.6x + 4 when x = −6

5. 9.2x − 3 when x = −1

6. Points O and P are shown.

a. Draw and label the image of point P under a dilation with center O and scale factor  2 . b. The distance between O and P is 6 cm. What is the distance between O and Pʹ ?

7. A sequence of transformations maps a figure onto its image. Which transformations could be part of a sequence that would always result in a figure that is congruent to its image? Choose all that apply. A. Dilation B. Reflection C. Rotation D. Translation

P ractice

EM2_0803SE_B_L06_daily_practice.indd 86

10/10/2020 7:08:54 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 7

LESSON Name

Date

Dilations on a Grid

1. Use any tools to draw and label the image of △ ABC under a dilation with center O and scale

factor  3 .

A B C

EM2_0803SE_B_L07_Classwork.indd 87

10/9/2020 4:04:00 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 7

Using a Grid to Measure 2. Plot and label the image of point D under a dilation with center O and scale factor 4.

3. Plot and label the image of point G under a dilation with center O and scale factor  3 .

EM2_0704TE_A_L01_g_3

L esson

EM2_0803SE_B_L07_Classwork.indd 88

10/9/2020 4:04:01 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 7

Using Properties of Dilations on a Grid

4. Plot and label the image of point B under a dilation with center O and scale factor  3 .

EM2_0803SE_B_L07_Classwork.indd 89

L esson

10/9/2020 4:04:01 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 7

Drawing Images of Polygons on a Grid

5. Draw and label the image of △ ABC under a dilation with center O and scale factor  3 .

A B C

O 6. Draw and label the image of △ FOX under a dilation with center O and scale factor  2 .

X O

Lesson

EM2_0803SE_B_L07_Classwork.indd 90

11/18/2020 11:26:33 AM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 7

EXIT TICKET Name

Date

Plot and label the image of point B under a dilation with center O and scale factor _  1 . 3

EM2_0803SE_B_L07_exit_ticket.indd 91

10/9/2020 4:05:16 PM

EM2_0803SE_B_L07_exit_ticket.indd 92

10/9/2020 4:05:16 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 7

RECAP Name

Date

Dilations on a Grid In this lesson, we •

plotted images of points under dilations on a grid.

•

drew images of polygons under dilations on a grid.

Examples

1. Plot and label the image of point P under a dilation with center O and scale factor _  3 . 2

Use OPʹ = r · OP to find the distance between O and Pʹ.

Pʹ Draw a ray, OP    . Count the units on the grid to find the distance between O and P.

⟶

OP = 8

EM2_0803SE_B_L07_lesson_recap.indd 93

OPʹ = _3 · 8 = 12

2 Plot Pʹ 12 units from O on the ⟶ ray, OP    .

10/9/2020 4:23:47 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 7

2. Plot and label the image of point Q under a dilation with center O and scale factor 3.

Finally, to locate Qʹ, draw ⟶ a segment from Rʹ to OQ    that is parallel to RQ   ¯.

Qʹ Q O

Rʹ

Plot R on the same vertical grid line as Q and the same horizontal grid line as O.

Then plot the image of R under the given dilation.

3. Draw and label the image of △PQR under a dilation with center O and scale factor _  2 . 3

Apply the strategy used in problem 2 to plot the image of each vertex of △PQR.

Q P Pʹ

O Sʹ S

R E CA P

EM2_0803SE_B_L07_lesson_recap.indd 94

Qʹ Rʹ

Tʹ Uʹ T

Connect the images of the vertices to draw △PʹQʹRʹ.

10/9/2020 4:23:48 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 7

PRACTICE Name

Date

1. A dilation with center O is shown. What is the scale factor of the dilation?

Gʹ

2. Plot and label the image of point C under a dilation with center O and scale factor _  4  . 5

EM2_0803SE_B_L07_daily_practice.indd 95

10/9/2020 4:32:44 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 7

3. Each diagram shows a dilation with center O. Find each scale factor and write it by the corresponding diagram. Choose from the following scale factors: _  2  ,  _   , 2, and  _2  . 4 1 3

Aʹ

Aʹ O

Aʹ

P ractice

EM2_0803SE_B_L07_daily_practice.indd 96

A Aʹ

10/9/2020 4:32:44 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 7

For problems 4–7, plot and label the image of point S under a dilation with center O and the given scale factor. 1 5. _   2

4. 4

S O

5 6. _   3

7. 1

S O

EM2_0803SE_B_L07_daily_practice.indd 97

P ractice

10/9/2020 4:32:45 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 7

8. Draw and label the image of △PQR under a dilation with center O and scale factor _  3  . 2

P Q

O 9. Draw and label the image of figure STUV under a dilation with center O and scale factor _  3  . 8

P ractice

EM2_0803SE_B_L07_daily_practice.indd 98

10/9/2020 4:32:45 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 7

10. Shawn plots the image of point M under a dilation with center O and scale factor 0.6 as shown in the diagram.

Mʹ

Lʹ

Use the properties of dilations to explain how Shawn determines the location of point M ʹ.

EM2_0803SE_B_L07_daily_practice.indd 99

P r ac t i c e

10/19/2020 11:45:28 AM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 7

Remember For problems 11–14, evaluate the expression for the given value. 11. 0.4x + 5 when x = 0.8

12. 1.2x − 5 when x = 0.3

13. 0.6x + 4 when x = −1.4

14. 6.2x − 9 when x = −0.8

15. The diagram shows a dilation with center O and scale factor  3  .

Pʹ

Qʹ

¯. Draw and label the image of point R under a dilation with center O a. Place a point R on   PQ 7 _ and scale factor  3  . b. How do you know the image of point R is in the correct location?

100

P ractice

EM2_0803SE_B_L07_daily_practice.indd 100

10/9/2020 4:32:46 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 7

16. In the given diagram, two lines meet at a point that is also the endpoint of a ray.

D B

(5x – 2)° (3x + 4)°

F E

A a. In a complete sentence, describe an angle relationship that would help you solve for x.

b. Determine the measures of ∠AFB and ∠BFC.

EM2_0803SE_B_L07_daily_practice.indd 101

P ractice

101

10/9/2020 4:32:46 PM

EM2_0803SE_B_L07_daily_practice.indd 102

10/9/2020 4:32:46 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 8

LESSON Name

Date

Dilations on the Coordinate Plane ¯ under a dilation with center O and scale factor 2. 1. Draw and label the image of  JK 

K O

EM2_0803SE_B_L08_classwork.indd 103

103

10/9/2020 5:44:11 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 8

Finding a Pattern 2. Consider points O and B in the coordinate plane. y 20 19 18 17 16 15 14 13 12 11 10 9 8 7

6 5 4 3 2 1 0

O 1

9 10 11 12 13 14 15 16 17 18 19 20

a. Choose a Scale Factor card. Then plot and label the image of point B under a dilation with center O and scale factor r from the card. b. Check one another’s images. Then complete the table for the group’s four images. Point B Coordinates

Scale Factor

Point B′ Coordinates

c. Look for a pattern within each row of the table. What do you notice about the relationship between the scale factor and the coordinates of points B and B′?

104

LESSON

EM2_0803SE_B_L08_classwork.indd 104

10/9/2020 5:44:11 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 8

Formalizing a Pattern 3. Use the pattern you found in problem 2 to complete the table for dilations centered at the origin. The first row is completed for you. Point P Coordinates

Scale Factor

Multiply

Point P′ Coordinates

(3, 10)

(2 ⋅ 3, 2 ⋅ 10)

(6, 20)

(1, 3)

(15, 9)

(8, 16)

1   4

(x, y)

4. What are the coordinates of the image of a point (x, y) under a dilation centered at the origin with scale factor r?

EM2_0803SE_B_L08_classwork.indd 105

L esson

105

10/9/2020 5:44:11 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 8

For problems 5 and 6, plot the image of point K under a dilation centered at the origin with scale factor 2. •

Use the grid strategy from lesson 7 to plot and label the actual location of point K′.

•

Use the rule to calculate the coordinates of point K′.

•

Compare the actual location to the calculated location of point K′. Does the rule work? y

5. 10 9 8 7 6 5 4 3 2 1 −10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2 K −3 −4 −5 −6 −7 −8 −9 −10

106

L esson

EM2_0803SE_B_L08_classwork.indd 106

1 2 3 4 5 6 7 8 9 10

10 9 8 7 6 5 4 3 2 1 −10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10

1 2 3 4 5 6 7 8 9 10

10/9/2020 5:44:11 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 8

Applying a Pattern For problems 7 and 8, find the coordinates of the image of each vertex under a dilation centered at the origin with the given scale factor r. Then graph and label the image of the figure. 7. r = 2.5 y 4

Q −16

−14

−12

−10

–8

–6

–4

–2

R S

–4 –6 −8

−10 −12 −14 −16

EM2_0803SE_B_L08_classwork.indd 107

L esson

107

10/9/2020 5:44:12 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 8

1   8. r = _ 3 y 10 9 8

7 6 5 4 3 2

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

9 10

−2 −3 −4 −5

−6 −7

−8 −9 −10

108

Lesson

EM2_0803SE_B_L08_classwork.indd 108

10/20/2020 12:48:01 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 8

For problems 9 and 10, find the scale factor of the dilation centered at the origin. 9.

y 2 1 −2 −1 0 –1

–2

Dʹ

9 10 11 12 13 14 15 16 17 18

–3

Eʹ

–4 –5 –6 –7

Fʹ

–8 −9 −10

−11 −12 −13 −14 −15 −16

−17 −18

EM2_0803SE_B_L08_classwork.indd 109

Lesson

109

10/19/2020 12:00:43 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 8

10.

y 14

Cʹ

12 10 8 6

Bʹ

B −16

−14

−12

−10

–8

–6

–4

–2

Aʹ A

−2 −4

110

Lesson

EM2_0803SE_B_L08_classwork.indd 110

10/19/2020 12:01:13 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 8

EXIT TICKET Name

Date

Consider △BCD in the coordinate plane.

y 16 14 12 10 8 6 4 2

−16 −14 −12 −10 −8

−6

−4

−2

−4 −6

−8 −10 −12 −14 −16

a. Find the coordinates of the images of vertices B, C, and D under a dilation centered at the origin with scale factor 3. Explain how you found your answer.

b. Graph and label the image of △BCD under the same dilation.

EM2_0803SE_B_L08_exit_ticket.indd 111

111

10/9/2020 5:43:58 PM

EM2_0803SE_B_L08_exit_ticket.indd 112

10/9/2020 5:43:58 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 8

RECAP Name

Date

Dilations on the Coordinate Plane In this lesson, we •

applied dilations centered at the origin on the coordinate plane.

•

determined the scale factor of a dilation centered at the origin.

Examples

1. Graph and label the image of △ABC under a dilation centered at the origin with scale factor 1.5. y

A dilation centered at the origin with scale factor r maps a point with coordinates (x, y) to a point with coordinates (rx, ry).

10 9 8 7 6 5

Bʹ

3 2 1

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

A Aʹ

Cʹ

9 10

C −2 −3 −4 −5 −6 −7 −8 −9 −10

A (−6, −4) maps to A′(−9, −6) because 1.5 (−6) = −9 and 1.5 (−4) = −6. B (−4, 2) maps to B′(−6, 3) because 1.5 (−4) = −6 and 1.5 (2) = 3. C (−2, −2) maps to C′(−3, −3) because 1.5 (−2) = −3.

EM2_0803SE_B_L08_lesson_recap.indd 113

113

10/9/2020 6:02:51 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 8

2. Find the scale factor of the dilation centered at the origin. y 7

5 4

Identify the coordinates for a pair of corresponding points, such as A and A′.

Dʹ

2 1

Aʹ −6

−5

−4

−3

−2

−1

B Cʹ Bʹ 1

x 2

−1 −2 −3 −4

A(−3, 3) maps to A′(−1, 1). Let r represent the scale factor. The scale factor is  _1  . 3

114

R E CA P

EM2_0803SE_B_L08_lesson_recap.indd 114

r = −1 ___  = 1_  −3

Divide the image’s x-coordinate by the point’s x-coordinate, or divide the image’s y-coordinate by the point’s y-coordinate.

10/9/2020 6:02:52 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 8

PRACTICE Name

Date

1. Each point A is mapped to point A′ by a dilation centered at the origin with the given scale factor. Complete the table. Coordinates of A

Scale Factor

(−4, −2)

(6, −4)

1_   2

(−5, 3)

Coordinates of A′

2. Plot the image of point A under a dilation centered at the origin with scale factor 3. Write the coordinates of the image. y 10 9 8 7 6 5 4 3 2 1 −10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10

EM2_0803SE_B_L08_daily_practice.indd 115

1 2 3 4 5 6 7 8 9 10

115

10/10/2020 2:52:29 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 8

3. Plot the image of point Z under a dilation centered at the origin with scale factor _  3  . Write the 4 coordinates of the image. y 10 9 8 7 6 5 4 3 2 1 −10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10

1 2 3 4 5 6 7 8 9 10

4. Point A lies on the x-axis. A dilation centered at the origin maps point A to point A′. Which statements are true? Choose all that apply. A. Point A′ lies on the x-axis. B. Point A′ lies on the y-axis. C. Point A′ lies in Quadrant I. D. Point A′ has an x-coordinate of 0. E. Point A′ has a y-coordinate of 0.

116

P ractice

EM2_0803SE_B_L08_daily_practice.indd 116

10/10/2020 2:52:29 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 8

5. What are the coordinates of a point that maps back to itself under a dilation centered at the origin? Explain.

6. Consider △PQR in the coordinate plane.

Q P

y 10 9 8 7 6 5 4 R 3 2 1

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10

1 2 3 4 5 6 7 8 9 10

a. What are the coordinates of the images of vertices P, Q, and R under a dilation centered at the origin with scale factor 1.5?

b. Graph and label the image of △PQR under the same dilation.

EM2_0803SE_B_L08_daily_practice.indd 117

P ractice

117

10/10/2020 2:52:29 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 8

7. Graph and label the image of figure UVWZ under a dilation centered at the origin with scale factor  _1  . 2

y 10 9 8 7 6 5 4 3 2 1

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10

1 2 3 4 5 6 7 8 9 10

8. Graph and label the image of △RST under a dilation centered at the origin with scale factor 3. y

10 9 8 7 6 5 4 3 2 1 −10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10

118

P ractice

EM2_0803SE_B_L08_daily_practice.indd 118

2 3 4 5 6 7 8 9 10

10/10/2020 2:52:30 PM

EUREKA MATH2 8 ▸ M3 ▸ TB ▸ Lesson 8

For problems 9 and 10, find the scale factor of the dilation centered at the origin. y

Jʹ

10 9 8 7 6 5 4 3 2 1

K Kʹ Lʹ L

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10

Bʹ

Aʹ

Mʹ M

Cʹ A

10 9 8 7 6 5 4 B 3 2 1

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10

EM2_0803SE_B_L08_daily_practice.indd 119

10.

1 2 3 4 5 6 7 8 9 10

P ractice

119

10/10/2020 2:52:30 PM

EUREKA MATH2

8 ▸ M3 ▸ TB ▸ Lesson 8

Remember For problems 11–14, evaluate the expression for the given value. 11. 2(x + 4) when x = 6

12. 5(x + 9) when x = 3

13. −4(x + 8) when x = −8

14. −7(x − 3) when x = −5

15. Draw the image of figure AB under a dilation with center O and scale factor 2.

A B O

16. A small bottle of soap costs $8.00 for 40 fluid ounces. A large bottle of soap costs $19.20 for 120 fluid ounces. What is the unit cost for each bottle size?

120

P ractice

EM2_0803SE_B_L08_daily_practice.indd 120

10/10/2020 2:52:30 PM

Similar Figures

TOPIC

Similar Viewing Experiences Ugh. How can you watch a movie

It’s not so bad. In fact, this

... in what way?

When you want to watch a video, you have a dizzying number of choices—not just of what video to watch, but also at what size to watch it. Do you want a phone screen, which is roughly 5 inches along the diagonal?

EM2_0803SE_C_topic_opener.indd 121

121

10/10/2020 1:19:42 PM

8 ▸ M3 ▸ TC

EUREKA MATH2

Or a tablet, which is roughly 9 inches? Or perhaps a laptop, which is roughly 14 inches? Or a TV, which may be as large as 4 feet? Or even a huge movie theater screen, which is 90 feet or more? This raises another question. What allows us to view the same content on screens of such radically different sizes? It’s possible only because of a mathematical agreement we’ve all reached: We will build most of our screens and film many of our TV shows and movies with the same aspect ratio. That ratio happens to be 16 : 9. In other words, the video should be 16 units wide, and 9 units high, no matter how large or small those units are. That’s why movies on a phone screen don’t look smooshed but just look smaller.

122

TO P I C O P E N E R

EM2_0803SE_C_topic_opener.indd 122

10/10/2020 1:19:42 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 9

LESSON Name

Date

Describing Dilations Finding the Center

What Is the Pattern?

Map It!

EM2_0803SE_C_L09_classwork.indd 123

123

10/10/2020 1:23:41 PM

EM2_0803SE_C_L09_classwork.indd 124

10/10/2020 1:23:41 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 9

EXIT TICKET Name

Date

Figure A′B′C′D′ is the image of figure ABCD under a dilation.

Aʹ Cʹ Dʹ

B Bʹ

a. Locate the center of dilation. Label it O. b. What is the scale factor of this dilation? Explain what this scale factor means.

c. What is the center and scale factor of the dilation that maps figure A′B′C′D′ back onto figure ABCD?

EM2_0803SE_C_L09_exit_ticket.indd 125

125

10/30/2020 5:54:03 PM

EM2_0803SE_C_L09_exit_ticket.indd 126

10/10/2020 1:23:19 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 9

RECAP Name

Date

Describing Dilations In this lesson, we •

located the center of a dilation given a figure and its image under the dilation.

•

precisely described a dilation given a figure and its image by using the center and scale factor of the dilation.

•

identified that the scale factor of a dilation that maps a figure onto its image is the reciprocal of the scale factor of the dilation that maps the image back onto the figure.

Example Figure S′T′U′V′ is the image of figure STUV under a dilation. Use lines to connect corresponding points on the figure and its image.

Sʹ

Tʹ

Vʹ 2 Uʹ

The center of dilation is the point where all the lines intersect.

a. Locate the center of dilation. Label it O. b. Describe the dilation that maps figure STUV onto figure S′T′U′V′. A dilation with center O and scale factor _  1  maps 4 figure STUV onto figure S′T′U′V′.

EM2_0803SE_C_L09_lesson_recap.indd 127

The scale factor of the dilation that maps figure STUV onto figure S′T′U′V′ is the reciprocal of the scale factor of the dilation that maps figure S′T′U′V′ onto figure STUV.

127

10/10/2020 4:27:26 PM

EM2_0803SE_C_L09_lesson_recap.indd 128

10/10/2020 4:27:26 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 9

PRACTICE Name

Date

1. A dilation maps △EFG onto △E′F′G′. Locate the center of dilation. Label it O.

Eʹ G

Gʹ Fʹ

2. A dilation maps figure JKLMN onto figure J′K′L′M′N′. Locate the center of dilation. Label it O.

Mʹ

Nʹ Jʹ N

M Kʹ

J K

EM2_0803SE_C_L09_daily_practice.indd 129

Lʹ

129

10/10/2020 1:23:31 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 9

¯ onto   ¯ 3. A dilation maps  PQ  P′Q′ .

P Pʹ 24

16 Q

Qʹ

a. Locate the center of dilation. Label it O. ¯ onto   ¯ P′Q′ . b. Describe the dilation that maps  PQ 

4. Consider the diagram.

C B

15 Bʹ

D Aʹ

Cʹ O Dʹ

a. Describe the dilation that maps figure ABCD onto figure A′B′C′D′.

b. Describe the dilation that maps figure A′B′C′D′ back onto figure ABCD.

130

P ractice

EM2_0803SE_C_L09_daily_practice.indd 130

10/10/2020 1:23:31 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 9

5. Point P′ is the image of point P under a dilation. Complete the table by writing the scale factor needed to apply each dilation. Scale Factor of the Dilation Maps P to P′

Maps P′ to P

_2   3

_5   4 1_ 6

_9   8 ¯. What is the scale factor of the dilation that ¯ onto   J′K′  6. A dilation with scale factor r maps   JK  ¯ ¯ maps   J′K′  back onto  JK ? Explain.

7. A dilation with scale factor __  a  maps figure RSTU onto figure R′S′T′U′. What is the scale factor b

of the dilation that maps figure R′S′T′U′ back onto figure RSTU ? Explain.

EM2_0803SE_C_L09_daily_practice.indd 131

P ractice

131

10/10/2020 1:23:31 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 9

8. A dilation with center O maps point N to point N′. What is the scale factor of the dilation that maps N′ back to N?

Nʹ

2 N 3 4

9. A dilation maps △STU onto △S′T′U′.

S Sʹ

55.5 44.4 U

Tʹ

Uʹ

a. Locate the center of dilation. Label it O.

b. Describe the dilation that maps △STU onto △S′T′U′.

132

P ractice

EM2_0803SE_C_L09_daily_practice.indd 132

10/10/2020 1:23:32 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 9

10. A dilation maps figure STUV onto figure S′T′U′V′.

U 5 T Tʹ

1.5

Uʹ V Vʹ

Sʹ

S a. What is the center of dilation?

b. Describe the dilation that maps figure STUV onto figure S′T′U′V′.

EM2_0803SE_C_L09_daily_practice.indd 133

P ractice

133

10/10/2020 1:23:32 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 9

11. A dilation maps figure LMNPQR onto figure L′M′N′P′Q′R′.

Mʹ M Lʹ

Nʹ

1 9 10

P R Rʹ

Pʹ

Q Qʹ

a. Locate the center of dilation. Label it O. b. Describe the dilation that maps figure LMNPQR onto figure L′M′N′P′Q′R′.

c. Describe the dilation that maps figure L′M′N′P′Q′R′ back onto figure LMNPQR.

134

P ractice

EM2_0803SE_C_L09_daily_practice.indd 134

10/10/2020 1:23:32 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 9

12. A dilation maps figure EFGH onto figure E′F′G′H′. E

Eʹ

H Hʹ

F Fʹ

Gʹ

a. Locate the center of dilation. Label it O. b. What is the scale factor of this dilation? Explain how you know.

Remember For problems 13–16, evaluate the expression for the given value. 3 when x = 1_ 13. 5(x +  _ 4) 4

14. 7(x + 0.2) when x = 0.5

15. −2(x − 3_) when x = −1_ 5 5

16. −3(x − 1.3) when x = −0.2

EM2_0803SE_C_L09_daily_practice.indd 135

P ractice

135

10/10/2020 1:23:33 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 9

17. Consider the diagram.

B O

A C

a. Draw and label the image of △ABC under a dilation with center O and scale factor 4.

b. Compare the distance between O and A to the distance between O and A′.

¯ parallel to  ¯ A′B′ ? Explain. c. Is   AB 

18. A 10-foot ladder leans against a wall. The bottom of the ladder is 40 inches from the wall. What height does the top of the ladder reach on the wall? Round your answer to the nearest tenth of an inch. A. 126.5 inches B. 113.1 inches C. 41.2 inches D. 38.7 inches

136

P ractice

EM2_0803SE_C_L09_daily_practice.indd 136

10/10/2020 1:23:33 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 10

LESSON Name

Date

Sequencing Transformations 1. Logan says that a dilation maps figure ABCDE onto figure A′B′C′D′E′. Do you agree with Logan? Explain.

B C

Eʹ Dʹ Cʹ Aʹ

EM2_0803SE_C_L10_classwork.indd 137

E Bʹ

137

10/10/2020 4:29:06 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 10

Create Your Own Sequence For problems 2–7, flip a coin two times to create a sequence of transformations. After each coin flip, mark whether the coin lands heads or tails. •

Flip 1: Circle the transformation from the first column that matches your coin flip.

•

Flip 2: Circle the transformation from the second column that matches your coin flip.

2. Draw and label the image of point P under the following sequence of transformations. First

Second

Heads

Dilation with center O and scale factor 3

Heads

90° clockwise rotation around point O

Tails

Dilation with center O and _1 scale factor  2

Tails

90° counterclockwise rotation around point O

138

Lesson

EM2_0803SE_C_L10_classwork.indd 138

10/31/2020 3:38:27 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 10

¯ under the following sequence of transformations. 3. Draw and label the image of  ST  First

Second

Heads

⟶ Translation along  AB 

Heads

Dilation with center O and scale factor 2

Tails

⟶ Translation along  CD 

Tails

Dilation with center O and scale _1 factor  3 

A C T B D

EM2_0803SE_C_L10_classwork.indd 139

Lesson

139

10/31/2020 3:39:08 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 10

4. Draw and label the image of trapezoid ABCD under the following sequence of transformations. First

Second

Heads

Dilation with center O and scale factor 4

Heads

Tails

Dilation with center O and scale factor 3

Tails

Reflection across line 𝓂 Reflection across line 𝓃

𝓂

𝓃

D B

C O

140

Lesson

EM2_0803SE_C_L10_classwork.indd 140

10/31/2020 3:39:48 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 10

5. Plot and label the image of point A under the following sequence of transformations. First

Second

Heads

Translation 3 units down and 7 units right

Heads

Dilation centered at the origin with scale factor 1.5

Tails

Translation 5 units down and 1 unit left

Tails

Dilation centered at the origin with scale factor 2

y 10 9 8 7 6 5

4 3 2 1

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

−2 −3 −4 −5 −6 −7 −8 −9 −10

EM2_0803SE_C_L10_classwork.indd 141

Lesson

141

10/30/2020 5:57:11 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 10

6. Graph and label the image of △ABC under the following sequence of transformations. First

Second

Heads

Dilation centered at the origin _5 with scale factor  4

Heads

Reflection across the x-axis

Tails

Dilation centered at the origin _1 with scale factor  2

Tails

Reflection across the y-axis

y 10

9 8

7 6 5

4 3 2 1

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

−2 −3 −4 −5 −6 −7 −8 −9 −10

142

Lesson

EM2_0803SE_C_L10_classwork.indd 142

10/30/2020 11:59:03 AM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 10

7. Graph and label the image of figure ABCD under the following sequence of transformations. First Heads Tails

Second

180° rotation around the origin 90° counterclockwise rotation around the origin

Heads

Dilation centered at the origin _3 with scale factor  2

Tails

Dilation centered at the origin with scale factor 0.5

y 10 9 8 7 6 5 4 3 2 1

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

−2 −3 −4 −5

−6 −7 −8 −9 −10

EM2_0803SE_C_L10_classwork.indd 143

Lesson

143

10/30/2020 11:59:19 AM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 10

⟶ ¯, point O, and  AB  8. Consider   XY  .

Y B A

¯ under the following sequence of transformations. a. Draw and label the image of  XY  ⟶ • Translation along  AB  •

Dilation with center O and scale factor 3

¯ onto   ¯ b. Locate the center of dilation for the single dilation that maps  XY  X″Y″ . Label the point P. ¯ onto   ¯ X″Y″ ? c. What is the scale factor of the dilation with center P that maps  XY 

144

L esson

EM2_0803SE_C_L10_classwork.indd 144

10/10/2020 4:29:07 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 10

⟶ 9. Consider △MNO and  AB .

B O

a. Draw and label the image of △MNO under the following sequence of transformations. •

•

Dilation with center O and scale factor  2 ⟶ Translation along  AB 

b. Locate the center of dilation for the single dilation that maps △MNO onto △M″N″O″. Label the point P. c. What is the scale factor of the dilation with center P that maps △MNO onto △M″N″O″?

EM2_0803SE_C_L10_classwork.indd 145

L esson

145

10/10/2020 4:29:07 PM

EM2_0803SE_C_L10_classwork.indd 146

10/10/2020 4:29:07 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 10

EXIT TICKET Name

Date

Graph and label the image of figure JKLM under the following sequence of transformations. • •

Dilation centered at the origin with scale factor _  1 2

Reflection across the y-axis

y 10 9 8

7 6 5

4 3

2 1

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

9 10

−2 −3 −4 −5 −6 −7 −8 −9 −10

EM2_0803SE_C_L10_exit_ticket.indd 147

147

10/10/2020 1:20:56 PM

EM2_0803SE_C_L10_exit_ticket.indd 148

10/10/2020 1:20:56 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 10

RECAP Name

Date

Sequencing Transformations In this lesson, we •

applied sequences of transformations.

•

recognized a sequence that involves a dilation and a translation as a single dilation.

Examples

1. Draw and label the image of △EFG under the following sequence of transformations. •

180° rotation around the origin

•

Dilation centered at the origin with scale factor 2 y 10

Gʺ

9 8 7 6

F′(−3, 1) maps to F″(−6, 2) because 2(− 3) = − 6 and 2(1) = 2.

Eʺ

Gʹ

5 4 3

Fʺ

Eʹ Fʹ

2 1

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2

−3 −4 −5

−6 −7 −8 −9 −10

EM2_0803SE_C_L10_lesson_recap.indd 149

149

10/10/2020 1:20:47 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 10

⟶

2. Consider figure HIJK and  LM . Draw lines to connect the corresponding vertices of figure HIJK and figure H″I″J″K″.

Jʹ

Iʹ Hʹ

Kʹ

H The center of dilation is at the intersection point of all the lines.

K Jʺ

Iʺ Hʺ

Kʺ O

a. Draw and label the image of figure HIJK under the following sequence of transformations. • •

 1 Dilation with center K and scale factor _ 3 ⟶ Translation along  LM 

b. Locate the center of dilation for the single dilation that maps figure HIJK onto figure H″I″J″K″. Label the point O. c. What is the scale factor of the dilation with center O that maps figure HIJK onto figure H″I″J″K″? 1_ 3 The dilation in the sequence that maps HIJK to H′I′J′K′ and the dilation that maps HIJK to H″I″J″K″ share the same scale factor but have different centers.

150

R ecap

EM2_0803SE_C_L10_lesson_recap.indd 150

10/10/2020 1:20:47 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 10

PRACTICE Name

Date

1. Draw and label the image of point P under the following sequence of transformations. •

Dilation with center O and scale factor 4

•

Translation along  AB 

⟶

B 2. Plot and label the image of point T under the following sequence of transformations. • •

90° clockwise rotation around the origin

Dilation centered at the origin with scale factor _  1  2

y 10 9 8 7 6 5 4 3 2 1 −10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10

EM2_0803SE_C_L10_daily_practice.indd 151

T 1 2 3 4 5 6 7 8 9 10

151

10/10/2020 1:21:05 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 10

¯ under the following sequence of transformations. 3. Graph and label the image of  AB  •

Reflection across the y-axis

•

Dilation centered at the origin with scale factor 3 y 10 9 8 7 6 5 4 3 2 1 −10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 A −2 B −3 −4 −5 −6 −7 −8 −9 −10

1 2 3 4 5 6 7 8 9 10

¯ under the following sequence of transformations. 4. Draw and label the image of  ST  • •

270° counterclockwise rotation around point S

Dilation with center S and scale factor _  5  2

152

P ractice

EM2_0803SE_C_L10_daily_practice.indd 152

10/10/2020 1:21:05 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 10

5. Draw and label the image of figure EFGH under the following sequence of transformations. •

Dilation with center O and scale factor 2

•

Reflection across line ℓ 𝓁

E H

F O

6. Graph and label the image of figure QRSTU under the following sequence of transformations. • •

Rotation 90° clockwise around the origin

Dilation centered at the origin with scale factor  _1  2

y 10 9 8 7 6 5 4 3 2 1 −10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 R −2 −3 Q −4 −5 S −6 −7 −8 U T −9 −10

EM2_0803SE_C_L10_daily_practice.indd 153

1 2 3 4 5 6 7 8 9 10

P ractice

153

10/10/2020 1:21:06 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 10

7. Draw and label the image of figure MNOP under the following sequence of transformations. • • •

 2  Dilation with center O and scale factor _ 3

Reflection across line ℓ ⟶ Translation along  JK 

𝓁

J K

154

P ractice

EM2_0803SE_C_L10_daily_practice.indd 154

10/10/2020 1:21:06 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 10

⟶ 8. Consider △ JKL and  AB .

A B

a. Draw and label the image of △ JKL under the following sequence of transformations. •

•

 3  Dilation with center K and scale factor _ 4 ⟶ Translation along   AB 

b. Locate the center of dilation for the single dilation that maps △ JKL onto △ J″K″L″. Label the point O. c. What is the scale factor of the dilation with center O that maps △ JKL onto △ J″K″L″?

Remember For problems 9–12, solve for x. 9. 5x + 7 = 17

11. 15x + 28 = −2

EM2_0803SE_C_L10_daily_practice.indd 155

10. 8x − 5 = 83

12. 11x – 3 = −14

P r ac t i c e

155

10/31/2020 11:31:26 AM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 10

13. Figure BCDE and point O are shown. O

C D

a. Draw and label the image of figure BCDE under a dilation with center O and scale factor _  1  . 4

b. Compare the distance between O and C to the distance between O and C′.

¯ and   ¯ C′D′ ? c. Based on the properties of dilations, what is true about  CD 

¯? 14. What is the length of  AB  y 5 4

3 2 1

−5

−4 −3 −2 −1

0 −1 −2

−3 −4 −5

156

P ractice

EM2_0803SE_C_L10_daily_practice.indd 156

10/10/2020 1:21:07 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 11

LESSON Name

Date

Similar Figures 1. Which set of figures does not belong?

A B

Aʹ

D Bʹ

R S

T Tʹ

Sʹ

Dʹ

Qʹ

Rʹ

Cʹ Set 1

Set 2

Fʹ E

Eʹ Dʹ

F Set 3

EM2_0803SE_C_L11_Classwork.indd 157

Lʹ

Jʹ

L K Kʹ Set 4

157

10/30/2020 5:44:55 AM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 11

Describing a Sequence For problems 2–6, describe a sequence of rigid motions or dilations, or both, that shows one figure is similar to the other. Draw and label any necessary vectors, lines of reflection, centers of rotation, or centers of dilation.

2. △ ABC ∼ △ ADE

A B

2 units

8 units

3. △MAT ∼ △PIN

M P I

158

L esson

EM2_0803SE_C_L11_Classwork.indd 158

10/10/2020 1:18:41 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 11

4. Trapezoid LMNO ∼ trapezoid PQRS

S P

M R

EM2_0803SE_C_L11_Classwork.indd 159

L esson

159

10/10/2020 1:18:41 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 11

5. △DIG ∼ △FAN

15 14 13 12 11 10 9

D N

8 7 6

5 4

3 2 1

−15 −14 −13 −12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

−2 −3 −4 −5

6. Figure RAIN ∼ figure COLD

C A

D L

160

L esson

EM2_0803SE_C_L11_Classwork.indd 160

10/10/2020 1:18:41 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 11

Properties of Similar Figures

7. In the diagram, figure ABCDEF ∼ figure GHIJKL. y

10 9 8 7

3 2 1 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

F 1

E 2

9 10

−2 −3 −4 −5

−6 −7 −8

−9 −10

a. Find the value of the ratio of each side length in figure GHIJKL to its corresponding side length in figure ABCDEF. What do you notice?

b. Compare the measure of each angle in figure GHIJKL to the measure of its corresponding angle in figure ABCDEF. What do you notice?

EM2_0803SE_C_L11_Classwork.indd 161

L esson

161

10/10/2020 1:18:42 PM

EM2_0803SE_C_L11_Classwork.indd 162

10/10/2020 1:18:42 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 11

EXIT TICKET Name

Date

In the diagram, figure ABCD ∼ figure QRST. Describe a sequence of rigid motions or dilations, or both, that maps figure ABCD onto figure QRST. Draw and label any necessary vectors, lines of reflection, centers of rotation, or centers of dilation.

A B

T S

R Q

EM2_0803SE_C_L11_exit_ticket.indd 163

163

10/10/2020 1:18:16 PM

EM2_0803SE_C_L11_exit_ticket.indd 164

10/10/2020 1:18:16 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 11

RECAP Name

Date

Similar Figures In this lesson, we •

described a sequence of rigid motions or dilations, or both, to show that two figures are similar.

•

identified that in similar figures ▸ ▸

corresponding side lengths are proportional, and corresponding angles are congruent.

Terminology One figure is similar to another if there is a sequence of rigid motions or dilations, or both, that maps one figure onto the other.

Examples

1. Describe a sequence of rigid motions or dilations, or both, that shows figure PQRS ∼ figure VUTS. ℓ Draw and label any necessary vectors, lines, or points.

U T

5 cm For a sequence that maps figure PQRS onto figure VUTS, the scale factor of the dilation is ___  VS .

___ __

VS 10 = = 2 5 PS

To show that the figures are similar, find a sequence that maps figure PQRS onto figure VUTS or find a sequence that maps figure VUTS onto figure PQRS.

10 cm

A reflection across line ℓ followed by a dilation with center S and scale factor 2 maps figure PQRS onto figure VUTS.

EM2_0803SE_C_L11_lesson_recap.indd 165

165

10/30/2020 5:47:17 AM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 11

2. Describe a sequence of rigid motions or dilations, or both, that shows △ ABC ∼ △LMN. y

10 9 8 7 6

Use units or grid lines to determine the scale factor of the dilation.

MN 1 ___ = _ 5 BC

Aʹ

Cʹ

2 1

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2

Bʹ

M 3

−3 −4 −5 −6 −7 −8 −9 −10

A dilation centered at the origin with scale factor 1_  followed by a translation 4 units right and 5 1 unit down maps △ ABC onto △LMN.

166

R ecap

EM2_0803SE_C_L11_lesson_recap.indd 166

10/10/2020 4:35:26 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 11

PRACTICE Name

Date

1. A sequence of rigid motions or dilations, or both, maps △CAT onto △DOG. a. What is the relationship between △CAT and △DOG?

b. What is true about the corresponding angles and corresponding side lengths of △CAT and △DOG? 2. Which sequence shows △RST ∼ △UVW ?

V W U

P S

⟶

A. A translation along  OP  followed by a dilation with center O and scale factor 4 maps △RST onto △UVW.

B. A translation along  OP  followed by a dilation with center P and scale factor 4 maps △RST onto △UVW.

⟶

⟷

C. A dilation with center O and scale factor 4 followed by a reflection across  OP  maps △RST onto △UVW.

D. A dilation with center P and scale factor 4 followed by a reflection across  OP  maps △RST onto △UVW.

⟷

EM2_0803SE_C_L11_daily_practice.indd 167

167

10/10/2020 1:18:28 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 11

3. Which sequence shows △MNO ∼ △STU?

𝓁 N T S

M O

A. A dilation with center P and scale factor _  1followed by a reflection across line ℓ maps 2 △MNO onto △STU.

B. A dilation with center P and scale factor 2 followed by a reflection across line ℓ maps △MNO onto △STU.

C. A reflection across line ℓ followed by a dilation with center P and scale factor _  1maps 2 △MNO onto △STU.

D. A reflection across line ℓ followed by a dilation with center P and scale factor 2 maps △MNO onto △STU.

168

P ractice

EM2_0803SE_C_L11_daily_practice.indd 168

10/10/2020 1:18:28 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 11

For problems 4–10, describe a sequence of rigid motions or dilations, or both, that shows one figure is similar to the other. Draw and label any necessary vectors, lines of reflection, centers of rotation, or centers of dilation. 4. △FGH ∼ △EGI

G H 8 I

18 F

5. △DBE ∼ △ ABC

A 6 C B

E 12 D

EM2_0803SE_C_L11_daily_practice.indd 169

P ractice

169

10/10/2020 1:18:29 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 11

6. △FGH ∼ △CDE

E 5 D 10

G 7.5 H

7. △RST ∼ △MNO

7 6 5 4 3

−7 −6 −5 −4 −3 −2 −1 0 −1

N 1

9 10 11

−2 −3 −4 −5

−6 −7 −8 −9 −10 −11 −12

170

P ractice

EM2_0803SE_C_L11_daily_practice.indd 170

10/10/2020 1:18:29 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 11

8. Figure ABCDEFG ∼ figure HIJKLMN

B D

C H

1.4 E

0.7

9. Figure C ∼ figure D

y 15 14 13 12 11 10

9 8 7 6 5

4 3 2 1 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

−2 −3

EM2_0803SE_C_L11_daily_practice.indd 171

P ractice

171

10/10/2020 1:18:29 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 11

10. Figure ABCDE ∼ figure PQRST

y 10 9 8 7

4 3 2 1

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

9 10

−2 −3 −4

R Q

−5 −6 −7 −8 −9 −10

11. The sequence of transformations that maps one triangle onto a similar triangle includes a dilation with scale factor 2. •

Logan states that the side lengths of the larger triangle are twice as long as the side lengths of the smaller triangle.

•

Shawn states that the angle measures of the larger triangle are twice as large as the angle measures of the smaller triangle.

Do you agree with Logan? Do you agree with Shawn? Explain.

172

P ractice

EM2_0803SE_C_L11_daily_practice.indd 172

10/10/2020 1:18:29 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 11

Remember For problems 12–15, solve for x.

  x + 6 = 4 12. 1_ 3

13. 1_ x − 4 = 10 5

1 14. _ x + 7 = −7 3

15. __   1 x − 11 = −2 10

16. Plot and label the image of point A under a dilation with center O and scale factor _  1 . 4

17. Solve for a in the equation 0.4a − 8 = 20.8.

EM2_0803SE_C_L11_daily_practice.indd 173

P ractice

173

10/10/2020 1:18:30 PM

EM2_0803SE_C_L11_daily_practice.indd 174

10/10/2020 1:18:30 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 12

LESSON Name

Date

Exploring Angles in Similar Triangles

1. Circle the transformations you would use in a sequence to show that △DEF ∼ △ ABC. rotation

reflection

translation

dilation

F 7 cm

D A

10 cm

Triangle Angles

2. Follow the directions to draw two triangles, △ ABC and △DEF. Then decide whether they are similar. Complete part (a) with a partner. Then complete parts (b)–(f) independently. a. Choose two angle measures that will be in both triangles. Measure of angle 1: Measure of angle 2: b. Use a protractor to draw two distinct triangles with the angle measures from part (a) on the index card. Cut out the triangles.

EM2_0803SE_C_L12_Classwork.indd 175

175

11/18/2020 11:42:27 AM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 12

c. Trace both triangles onto the space provided. • • •

176

Label one △ ABC and the other △DEF.

Label the angles of each triangle with the measures you used from part (a). Measure the lengths of the sides and show that the lengths of the corresponding sides of the triangles are proportional.

L esson

EM2_0803SE_C_L12_Classwork.indd 176

10/10/2020 4:31:46 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 12

d. Lay the triangle cutouts on top of the traced triangles in the book. Is there a sequence of rigid motions that maps one of the angles in △ ABC onto a congruent angle in △DEF? If so, describe the rigid motions used. e. After applying a sequence of rigid motions, can a dilation that maps △ ABC onto △DEF be precisely described? If so, describe the dilation. f.

Is △ ABC similar to △DEF? Explain your reasoning.

Are They Similar? 3. Points A, K, and T are collinear, and points A, S, and C are collinear.

Are △ ASK and △ ACT similar? Complete parts (a)–(d) to determine the answer.

a. What two pairs of angles are congruent according to the diagram? Write the statements of congruence.

EM2_0803SE_C_L12_Classwork.indd 177

Lesson

177

10/30/2020 5:52:34 AM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 12

¯ ? Explain your conclusion. ¯ and   TC  b. Based on the information, what can we conclude about  KS 

c. Describe the sequence that maps △ ASK onto △ ACT.

d. What can you now conclude about △ ASK and △ ACT?

Angle–Angle Criterion For problems 4–9, determine whether the triangles are similar by the angle–angle criterion. Explain your reasoning. 4. K

25°

5. T 34°

80°

129°

25°

129° 34°

80° B

178

L esson

EM2_0803SE_C_L12_Classwork.indd 178

10/10/2020 4:31:47 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 12

6. T

129° A H

34°

32°

53° L

129°

17°

28°

M D

80° A

8. E

9. C

59°

31°

7 cm

D A

10 cm

31°

103°

U 31°

EM2_0803SE_C_L12_Classwork.indd 179

85°

L esson

179

10/10/2020 4:31:48 PM

EM2_0803SE_C_L12_Classwork.indd 180

10/10/2020 4:31:48 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 12

EXIT TICKET Name

Date

Determine whether the triangles are similar by the angle–angle criterion. Explain your reasoning.

B 55° A

55°

2. C

52° 70°

EM2_0803SE_C_L12_exit_ticket.indd 181

48°

70°

181

10/30/2020 5:53:45 AM

EM2_0803SE_C_L12_exit_ticket.indd 182

10/10/2020 1:19:02 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 12

RECAP Name

Date

Exploring Angles in Similar Triangles In this lesson, we •

recognized that triangles with two pairs of corresponding angles are similar.

•

defined the angle–angle criterion.

Angle–Angle Criterion If two triangles have two pairs of congruent angles, then the triangles are similar.

C P

∠ A ≅ ∠P and ∠B ≅ ∠Q, so △ ABC ∼ △PQR by the angle–angle (AA) criterion.

EM2_0803SE_C_L12_lesson_recap.indd 183

183

10/10/2020 1:18:53 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 12

Examples

Determine whether △ONE is similar to △SIX by the angle–angle criterion. Explain your reasoning. 1.

m∠S = 60° because

30°

N ∠N ≅ ∠I because both

the sum of the interior angle measures of a triangle is 180°.

60° 60°

are right angles.

30° X

△ONE ∼ △SIX by the angle–angle criterion because they have two pairs of congruent angles,

∠N ≅ ∠I and ∠O ≅ ∠S. 2.

E 63°

m∠O = 63° because the sum of the interior angle measures of a triangle is 180°. So ∠O is not congruent to ∠S.

63° I 54° 54° 73° N

53° X

△ONE is not similar to △SIX because they do not have two pairs of congruent angles.

184

R ecap

EM2_0803SE_C_L12_lesson_recap.indd 184

10/10/2020 1:18:53 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 12

PRACTICE Name

Date

For problems 1–5, •

state whether the two triangles are similar by the angle–angle criterion and explain your reasoning, and

•

if there is not enough information to determine whether the triangles are similar by the angle–angle criterion, explain why not.

50°

30°

EM2_0803SE_C_L12_daily_practice.indd 185

185

10/10/2020 4:33:07 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 12

40°

100°

47°

39°

94°

186

P ractice

EM2_0803SE_C_L12_daily_practice.indd 186

10/10/2020 4:33:08 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 12

For problems 6–9, determine whether △ ABC is similar to △ XYZ by the angle–angle criterion. Explain how you know.

50°

60° 87° C

60°

29°

Z B

40°

60°

9. A

79° X

55°

A 85° 85° X

EM2_0803SE_C_L12_daily_practice.indd 187

46°

45° C

55°

P r ac t i c e

187

10/30/2020 5:56:13 AM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 12

10. Is △NOP similar to △LOM by the angle–angle criterion? Explain.

35°

80° P

M 65° L

188

P ractice

EM2_0803SE_C_L12_daily_practice.indd 188

10/10/2020 4:33:09 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 12

¯ onto   ¯ 11. The diagram shows a dilation with center A and scale factor 1.5 that maps  BC  BʹCʹ . A 45° B

C Bʹ

Cʹ

Henry and Sara are asked to explain how they know that △ABC ∼ △ABʹCʹ. Henry’s Response

I know that △ ABC ∼ △ ABʹCʹ because ¯ onto   ¯ BʹCʹ  . a dilation is shown that maps  BC  △ That means the same dilation maps ABC onto △ ABʹCʹ, which means △ ABC ∼ △ ABʹCʹ.

Sara’s Response

I know that △ ABC ∼ △ ABʹCʹ because I can use parallel lines and corresponding angles to show that ∠ ABC ≅ ∠ ABʹCʹ and ∠ ACB ≅ ∠ ACʹBʹ. By the angle–angle criterion, △ ABC ∼ △ ABʹCʹ.

Is Henry’s response correct? Is Sara’s response correct? Explain.

Remember For problems 12–15, solve for x. 12. 0.5x + 7 = 8.5

13. 0.4x − 8 = 13.2

14. 0.2x + 4 = −11.2

15. 1.2x − 1.3 = −8.2

EM2_0803SE_C_L12_daily_practice.indd 189

P ractice

189

10/10/2020 4:33:09 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 12

16. Plot and label the image of point A under a dilation with center O and scale factor 3.

A O 17. The table shows the total cost in dollars to buy muffins. Write an equation to represent the relationship between the total cost t and the number of muffins m.

190

Number of Muffins

Total Cost (dollars)

8.70

11.60

20.30

P r act i c e

EM2_0803SE_C_L12_daily_practice.indd 190

10/10/2020 7:23:35 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 13

LESSON Name

Date

Similar Triangles Composition 8 by Wassily Kandinsky

What kind of feeling do you think the artist was trying to convey through this piece? Why?

EM2_0803SE_C_L13_classwork.indd 191

191

10/19/2020 12:43:00 PM

8 ▸ M3 ▸ TC ▸ Lesson 13

EUREKA MATH2

What do you notice about the shapes in the painting?

What do you wonder about the shapes in the painting?

192

L esson

EM2_0803SE_C_L13_classwork.indd 192

10/10/2020 1:20:31 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 13

For problems 1 and 2, a diagram of figures has been recreated from an enlarged section of the painting. Determine whether the recreated figures are similar.

1. Is quadrilateral QUIZ similar to quadrilateral 2. Is △MAT similar to △MUD? Explain. QUAD? Explain.

Z I

EM2_0803SE_C_L13_classwork.indd 193

D M U

Lesson

193

10/30/2020 10:39:01 AM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 13

Angle Relationships for Similarity 3. List as many angle relationships as possible. Angle Relationships

194

Lesson

EM2_0803SE_C_L13_classwork.indd 194

10/30/2020 5:57:12 AM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 13

4. Consider △KLM and △STU shown in the diagram. L

38° M

K U

S 127°

38°

a. Complete the table with the angle measures. Then identify the angle relationship that allowed you to find the measure. Angle in △ KLM

m∠MKL = m∠KLM = m∠KML = 38°

Corresponding Angle in △STU

Angle Relationship

m∠UST = 127° m∠STU = m∠SUT = 38°

Given

b. Is △KLM similar to △STU by the angle–angle criterion? Explain.

EM2_0803SE_C_L13_classwork.indd 195

L esson

195

10/10/2020 1:20:32 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 13

For problems 5–8, determine whether the given triangles are similar by the angle–angle criterion. Explain your reasoning. If there is not enough information to determine whether the triangles are similar by the angle–angle criterion, explain why not. 5. △MYB and △GAB

18°

B 117°

196

L esson

EM2_0803SE_C_L13_classwork.indd 196

10/10/2020 1:20:33 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 13

6. △FTI and △FHY

F 79° 54° T

46° H

EM2_0803SE_C_L13_classwork.indd 197

L esson

197

10/10/2020 1:20:33 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 13

7. △CUS and △BUA

S C U B

198

L esson

EM2_0803SE_C_L13_classwork.indd 198

10/10/2020 1:20:33 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 13

8. △GTH and △GEI

I E

EM2_0803SE_C_L13_classwork.indd 199

L esson

199

10/10/2020 1:20:33 PM

EM2_0803SE_C_L13_classwork.indd 200

10/10/2020 1:20:33 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 13

EXIT TICKET Name

Date

Is △ ABC similar to △KLM by the angle–angle criterion? Explain.

B 24° 136° C

K 24°

EM2_0803SE_C_L13_exit_ticket.indd 201

201

10/10/2020 1:20:02 PM

EM2_0803SE_C_L13_exit_ticket.indd 202

10/10/2020 1:20:02 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 13

RECAP Name

Date

Similar Triangles In this lesson, we •

used angle relationships to find unknown angle measures in diagrams.

•

determined whether two triangles are similar by the angle–angle criterion.

Examples

1. Determine whether △LJK is similar to △LHG by the angle–angle criterion. Explain your reasoning.

The sum of the interior angle measures of a triangle is 180°.

m∠LGH = 52° and m∠L JK = 81°.

50° H

78°

49°

△ LJK is not similar to △LHG. Because m∠LGH =

have only one pair of congruent angles.

EM2_0803SE_C_L13_lesson_recap.indd 203

K 52° and m∠LJK = 81°, the two triangles

203

10/30/2020 6:00:49 AM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 13

2. Determine whether △ JKL is similar to △NML. Explain your reasoning.

∠JKL and ∠NML are

alternate interior angles.

L M

△ JKL ∼ △ NML by the angle–angle criterion. ∠JLK ≅

∠JLK and ∠NLM are vertical angles.

∠NLM because they are vertical

angles. ∠JKL ≅ ∠NML because they are alternate interior angles created by parallel lines cut by a transversal.

204

R ecap

EM2_0803SE_C_L13_lesson_recap.indd 204

10/10/2020 1:19:52 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 13

PRACTICE Name

Date

For problems 1–4, determine whether △WIT is similar to △PIC by the angle–angle criterion. Explain your reasoning. 2. I

80° W

30°

40°

T 55°

P I

EM2_0803SE_C_L13_daily_practice.indd 205

P I

205

10/10/2020 1:20:15 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 13

5. Consider the diagram.

J L

I 34°

a. If the angle measures in △GHI and △ JKL can be found, list them in the table. Then identify the angle relationship that allowed you to find the measure. Angle in △GHI

Corresponding Angle in △ JKL

Angle Relationship

b. Is △GHI similar to △ JKL by the angle–angle criterion? Explain.

206

P r act i c e

EM2_0803SE_C_L13_daily_practice.indd 206

10/30/2020 6:02:44 AM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 13

6. Is △ ABC similar to △DEC by the angle–angle criterion? Explain.

24° 100° C E

24° D

For problems 7–9, •

state whether the two triangles are similar by the angle–angle criterion and explain your reasoning;

•

if there is not enough information to determine whether the triangles are similar by the angle–angle criterion, explain why not; and

•

use some or all of the following vocabulary terms in your explanations.

angle sum of a triangle

EM2_0803SE_C_L13_daily_practice.indd 207

vertical angles

alternate interior angles

right angle

linear pair

P ractice

207

10/10/2020 1:20:16 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 13

7. △TOY and △CON

Y C

O T

8. △WIZ and △TIX

Z W I

208

P ractice

EM2_0803SE_C_L13_daily_practice.indd 208

10/10/2020 1:20:16 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 13

37°

18°

a. △ NAH and △YAS

b. △NAS and △YAH

EM2_0803SE_C_L13_daily_practice.indd 209

P ractice

209

10/10/2020 1:20:17 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 13

10. One of the properties of a trapezoid is that the diagonals of the trapezoid form a pair of similar triangles.

K J

a. Select the pair of similar triangles. A. △ KLX and △ XWK B. △ JLX and △ JWK C. △ JKL and △ JXW

D. △LXW and △ LWK

b. Explain why the triangles you selected are similar by the angle–angle criterion.

210

P ractice

EM2_0803SE_C_L13_daily_practice.indd 210

10/10/2020 1:20:17 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 13

11. Ethan and Liam were given the following diagram and asked to explain why △ NEW ∼ △GEO by the angle–angle criterion.

93°

O 36°

G 51°

E Ethan’s First Step

Liam’s First Step

Used the angle sum of a triangle for △ NEW

Used the angle sum of a triangle for △GEO

Choose Ethan’s or Liam’s first step and continue with their reasoning to show that △ NEW ∼ △GEO by the angle–angle criterion.

EM2_0803SE_C_L13_daily_practice.indd 211

P ractice

211

10/10/2020 1:20:17 PM

EUREKA MATH2

8 ▸ M3 ▸ TC ▸ Lesson 13

Remember For problems 12–15, solve for x. 12. 5(x + 12) = 10

13. 4(x − 8) = 60

14. 6(x + 3) = −30

15. 10(x − 4) = −100

16. △ ABC is shown in the coordinate plane.

y 10 9 8 7 6 5 4 3 2 1

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

−2

9 10

−3 −4 −5

−6 −7 −8 −9 −10

a. Graph the image of △ ABC under a dilation centered at the origin with scale factor 2. Label the image △ AʹBʹCʹ.

b. What are the coordinates of the vertices of △ AʹBʹCʹ?

c. Describe how you located the coordinates of the vertices of △ AʹBʹCʹ.

212

P ractice

EM2_0803SE_C_L13_daily_practice.indd 212

10/10/2020 1:20:18 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 13

17. The student council sells tickets to the school dance. The graph shows the amount of money the student council earns by selling tickets on four different days. School Dance Ticket Sales

y 50 45 Money Earned (Dollars)

40 35 30 25 20 15 10 5 0

10 12 14 16 18 20 22 24

Number of Tickets Sold

a. What does the point (2, 5) represent?

b. What is the unit rate?

c. What point would you plot on the graph to indicate the unit rate?

d. What does the unit rate represent?

EM2_0803SE_C_L13_daily_practice.indd 213

P ractice

213

10/10/2020 1:20:18 PM

EM2_0803SE_C_L13_daily_practice.indd 214

10/10/2020 1:20:18 PM

Applications of Similar Figures

TOPIC

We're So Similar! You have a 30o So weird! Do you also have a 60o angle?

Wait... is your hypotenuse exactly twice as long as your shorter leg?? IT’S LIKE YOU READ MY MIND!

In mathematics, we use the word similar in a special and specific way. It doesn’t mean “looking pretty much alike.” It doesn’t mean “having a lot in common.” It doesn’t mean “Yeah, I can see the resemblance.” It means that the two figures match up under a sequence of rigid motions or dilations, or both. And that meaning has big consequences for what the two figures wind up looking like. You can expect not only the same angle measures, but side lengths with the same proportions, too—every part of one figure relates to every part of the other figure. That’s why, across mathematics, when you want to know more about a figure, it helps to identify the figure’s similar partner. Study that figure’s partner, and you’ll learn loads about the original figure.

EM2_0803SE_D_topic_opener.indd 215

215

10/10/2020 1:27:18 PM

EM2_0803SE_D_topic_opener.indd 216

10/10/2020 1:27:18 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 14

LESSON Name

Date

Using Similar Figures to Find Unknown Side Lengths 1. What strategies could you use to find the height of the rectangle?

Unknown Lengths in Similar Figures 2. △ ABC is similar to △ ADE.

B 15 C

5 A

¯? a. What is the length of  EA 

EM2_0803SE_D_L14_Classwork.indd 217

217

10/10/2020 5:56:33 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 14

¯? ¯ is 17.1 units, what is the length of  BC  b. If the length of  DE 

¯ and   AE  ¯. 3. In the diagram, △ APE ∼ △ ANT. Find the lengths of  AN 

P N 9

18.3

6 E

218

L esson

EM2_0803SE_D_L14_Classwork.indd 218

11.4

10/10/2020 5:56:33 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 14

4. A user pinches the touch screen of a tablet computer to reduce the size of a full-screen picture. The diagram shows the full screen and the reduced picture, which are similar figures. Find the screen length of the tablet computer in inches.

2.4 in 1.75 in

7 in

EM2_0803SE_D_L14_Classwork.indd 219

L esson

219

10/10/2020 5:56:33 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 14

5. In the diagram, △ELK ∼ △HUG and △HUG ∼ △WIN. Find all the unknown side lengths of the triangles.

E 12 L 13 9

H K

U 13.5 W

6.75

G N

220

L esson

EM2_0803SE_D_L14_Classwork.indd 220

10/10/2020 5:56:34 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 14

Finding Unknown Side Lengths ¯. 6. Find the length of  ET 

L 1.23

2.46 39°

E 123°

39°

3.3

O 6.6 T 5

18°

EM2_0803SE_D_L14_Classwork.indd 221

Lesson

221

10/30/2020 6:03:54 AM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 14

¯. 7. Find the length of  EK 

5.1

222

Lesson

EM2_0803SE_D_L14_Classwork.indd 222

10/30/2020 6:04:41 AM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 14

8. What is the height of the rectangle?

EM2_0803SE_D_L14_Classwork.indd 223

Lesson

223

10/30/2020 6:14:03 AM

EM2_0803SE_D_L14_Classwork.indd 224

10/10/2020 5:56:34 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 14

EXIT TICKET Name

Date

¯. Find the length of  AB  A 40°

D 11

EM2_0803SE_D_L14_exit_ticket.indd 225

30°

30° 110° 7.5 F

225

10/30/2020 6:15:07 AM

EM2_0803SE_D_L14_exit_ticket.indd 226

10/10/2020 1:26:25 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 14

RECAP Name

Date

Using Similar Figures to Find Unknown Side Lengths In this lesson, we •

assessed whether two triangles are similar.

•

used properties of similar figures to find unknown side lengths.

Examples ¯. 1. Find the length of  MO 

37° 6

78°

65°

O R

First, determine whether △MNO is similar to △PQR.

78°

△MNO ∼ △PQR by the angle–angle criterion. The sum of the interior measures of a triangle

is 180°, which means m∠MNO = 37°. So ∠OMN ≅ ∠RPQ and ∠MNO ≅ ∠PQR. MO MN ___   = ___   PR PQ

This type of equation is called a proportion.

MO 5_ ___   =   4 6

___ 4(MO ) = 4(5_ ) 4

¯ MO = 3.  3 ¯ is 3. ¯ The length of  MO   3 units.

EM2_0803SE_D_L14_lesson_recap.indd 227

Because the triangles are similar, write a proportion to represent the relationship between the lengths of the corresponding sides.

227

11/5/2020 12:01:06 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 14

¯ and   CA  ¯. 2. Find the lengths of  CD 

The triangles share ∠BCD.

C 5 B

¯ and AE    BD    ¯ are parallel segments with transversal CA    ¯.

7.2

11.6 D

A 14.4 E

△ ACE ∼ △BCD by the angle–angle criterion. The diagram shows that ∠ACE ≅

∠BCD. I know ∠EAC ≅ ∠DBC because they are corresponding angles created by parallel segments cut

by a transversal.

CA AE ___   = ___   CB BD

CD ___ ___   =  BD  CE AE

CA 14.4 ___   = ____ 

7.2 ____   CD   =  ____   11.6 14.4

11.6(____   CD ) = 11.6(____   7.2 ) 11.6

CD = 5.8 ¯ is 5.8 units. The length of  CD 

228

R ecap

EM2_0803SE_D_L14_lesson_recap.indd 228

14.4

7.2

___ ____ 5(CA ) = 5(14.4 ) 5

7.2

CA = 10 ¯ is 10 units. The length of  CA 

10/10/2020 1:26:16 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 14

PRACTICE Name

Date

¯. 1. △BRI is similar to △GHT. Find the length of  GT 

4.2

13.6

¯. 2. △SMR is similar to △SAT. Find the length of  ST 

3.6 4.5

S 4

EM2_0803SE_D_L14_daily_practice.indd 229

229

10/10/2020 1:26:36 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 14

3. Each diagram shows similar triangles. Find the length of the segment and write it by the corresponding diagram. Choose from the following segment lengths: 4.8 units, 7.5 units, 30 units, and 42 units. ¯? a. △ AYE is similar to △ ADR. What is the length of  AY 

15 6

A D

¯? b. △RAN is similar to △REL. What is the length of  EL 

L 15 N

230

P ractice

EM2_0803SE_D_L14_daily_practice.indd 230

12 6

10/10/2020 1:26:36 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 14

4. Figure ZERO is similar to figure LIST.

I 9.5

4.4

3.8 T

O 4.4

5 R

Z a. Find the scale factor of the dilation that maps figure ZERO onto figure LIST.

b. Find the length of  ¯ ZE .

¯. c. Find the length of  ST 

TL . d. Find the length of  ¯

EM2_0803SE_D_L14_daily_practice.indd 231

P ractice

231

10/10/2020 1:26:37 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 14

¯. If not, 5. Is △VET similar to △VOX by the angle–angle criterion? If so, find the length of  TV  explain how you know.

O 29°

E 2.1

29°

8.4 T

99°

4.2

232

P ractice

EM2_0803SE_D_L14_daily_practice.indd 232

10/10/2020 1:26:37 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 14

¯ and   AK  ¯. 6. Find the lengths of  ZA  P

4 Z

Y 4.8 2.7

¯. 7. Find the length of  RC  A M R 34°

8 56° C 4.5 I

EM2_0803SE_D_L14_daily_practice.indd 233

P ractice

233

10/10/2020 1:26:37 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 14

¯. 8. Find the length of  SL  L 70° O

16 9.6

6.6

40° 70°

40°

9.6 16

234

P ractice

EM2_0803SE_D_L14_daily_practice.indd 234

10/10/2020 1:26:37 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 14

¯,   IX  ¯, and   IT  ¯. 9. Find the lengths of  FI  I X

4.5

2 P

1.8

7.2

8.1

A F

EM2_0803SE_D_L14_daily_practice.indd 235

P ractice

235

10/10/2020 1:26:38 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 14

Remember For problems 10–13, solve for x.

1 (x + 8) = 6 10. _ 2

1 11. _ (x + 7) = 1 8

4 12. _ (x + 4) = 8 5

2 13. _ (x − 8) = −2 3

14. A dilation is shown in the diagram.

Cʹ Dʹ

Aʹ

D A

C B

Bʹ

a. Find the center of dilation. Label it O. b. What is the scale factor of the dilation? Explain how you know.

c. Describe the dilation that maps figure AʹBʹCʹDʹ back onto figure ABCD.

15. Alex earns $114 for 8 hours of work. He earns the same amount of money for each hour he works. Write an equation that represents the relationship between Alex’s total earnings a and the total number of hours he works h.

236

P ractice

EM2_0803SE_D_L14_daily_practice.indd 236

10/10/2020 1:26:38 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 15

LESSON Name

Date

Applications of Similar Figures 1. Study the artwork in each category. Category 1

Category 2

Papyrus of Ani

Blue Hole, Flood Waters, Little Miami River

Ani

Robert Duncanson

Egypt, 1250 BCE

United States, 1851

The Kiss

Tenochtitlan

Gustav Klimt

Diego Rivera

Austria, 1908

Mexico, 1945

What do you think the difference is between the artwork in category 1 and the artwork in category 2?

EM2_0803SE_D_L15_classwork.indd 237

237

10/10/2020 6:18:40 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 15

2. Study the given examples.

Example A

Example B

Study for Le Pont de L’Europe Gustave Caillebotte France, 1876 Example C

The Avenue at Middelharnis Meindert Hobbema 1689, Netherlands What is the purpose of the lines in each of the examples?

238

L esson

EM2_0803SE_D_L15_classwork.indd 238

10/10/2020 6:18:41 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 15

Mirror Height For problems 3–5, use the space provided to take notes and draw a diagram from the video. 3. What is the height of the building to the nearest foot?

EM2_0803SE_D_L15_classwork.indd 239

L esson

239

10/10/2020 6:18:41 PM

8 ▸ M3 ▸ TD ▸ Lesson 15

EUREKA MATH2

Less Than Perfect Pole 4. What is the length of the flagpole to the nearest tenth of a foot?

240

L esson

EM2_0803SE_D_L15_classwork.indd 240

10/10/2020 6:18:41 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 15

Henry’s Kite 5. To the nearest foot, how high above the ground is Henry’s kite?

EM2_0803SE_D_L15_classwork.indd 241

Lesson

241

10/31/2020 11:35:23 AM

EM2_0803SE_D_L15_classwork.indd 242

10/10/2020 6:18:41 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 15

EXIT TICKET Name

Date

Lily stands near a streetlamp. The light from the streetlamp causes her to cast a shadow. Lily is 5.5 feet tall. The shadow she casts is 3 feet long. Lily is 10 feet away from the streetlight.

5.5 ft 10 ft

3 ft

What is the height of the streetlamp? Round your answer to the nearest tenth of a foot.

EM2_0803SE_D_L15_exit_ticket.indd 243

243

10/10/2020 6:18:58 PM

EM2_0803SE_D_L15_exit_ticket.indd 244

10/10/2020 6:18:58 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 15

RECAP Name

Date

Applications of Similar Figures In this lesson, we •

examined examples of linear perspective in artwork and related it to dilations.

•

used properties of similar figures to solve problems.

Example When Nora stands 1 40feet from the base of a tree, the tip of her shadow meets the tip of the tree’s shadow. Nora is 4 .5feet tall, and her shadow is10feet long. How tall is the tree?

Draw and label a diagram that represents the situation. Label the unknown length you need to find with a variable.

The two triangles share an angle, and both also have a right angle.

x 4.5 ft 140 ft

10 ft

The distance from the base of the tree to the tip of Nora’s shadow is 150 ft.

The two triangles are similar by the angle–angle criterion because they share an angle and both triangles also have a right angle. 150 x   = ___  ___   10 4.5

___   x   = 4.5 150   4.5 ___

(4.5)

EM2_0803SE_D_L15_lesson_recap.indd 245

( 10 )

x = 67.5

245

10/10/2020 1:28:27 PM

EM2_0803SE_D_L15_lesson_recap.indd 246

10/10/2020 1:28:27 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 15

PRACTICE Name

Date

1. Draw the following elements of the picture. •

Vanishing point

•

Two vanishing lines

2. How is linear perspective related to dilations?

EM2_0803SE_D_L15_daily_practice.indd 247

247

10/10/2020 1:28:45 PM

8 ▸ M3 ▸ TD ▸ Lesson 15

EUREKA MATH2

3. Shawn sits outside his building during lunch. He notices that his seated height is 3.5 feet, and he casts a shadow that is 1.4 feet long. The building casts a shadow that is 72.8 feet long. Diagram not drawn to scale.

a. How tall is Shawn’s office building?

b. The average ceiling height of each floor of the building is 14 feet. How many floors are in the building?

248

P ractice

EM2_0803SE_D_L15_daily_practice.indd 248

10/10/2020 1:28:45 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 15

4. Jonas measures the height of his apartment’s balcony. He lays a mirror on the ground 11.8 horizontal feet from the edge of the balcony. Then he moves farther from his apartment so that he is 6 feet from the mirror. If he looks into the mirror, he can see the bottom edge of his balcony. Jonas’s eyes are 5.1 feet from the ground. Diagram not drawn to scale. How high is the bottom edge of his balcony? Use similar triangles to show your work.

Jonas’s balcony

EM2_0803SE_D_L15_daily_practice.indd 249

P r ac t i c e

249

10/19/2020 1:00:02 PM

8 ▸ M3 ▸ TD ▸ Lesson 15

EUREKA MATH2

5. The Leaning Tower of Pisa is a bell tower in Italy that is famous because it tilts at an angle. Diagram not drawn to scale.

The top of the tower is approximately 183 feet from the ground. At a particular time of day, the tower casts a shadow of 12.8 feet. A flagpole was placed vertically on the tower, which added an additional 2.1 feet to the shadow. To the nearest whole foot, how tall is the flagpole on top of the Leaning Tower of Pisa? Use similar triangles to show your work.

250

P r act i c e

EM2_0803SE_D_L15_daily_practice.indd 250

10/19/2020 1:00:03 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 15

6. Create a context for the diagram. Provide a question based on the unknown length in the diagram, and then answer that question.

210

4.5

6.3

Remember For problems 7–10, solve for x. 7. 0.5(x + 10) = 8.2

8. 0.3(x − 4) = 9.6

9. 1.2(x + 5) = −4.2

10. 6.4(x − 1) = −2.4

EM2_0803SE_D_L15_daily_practice.indd 251

P r ac t i c e

251

10/19/2020 1:00:03 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 15

11. Graph and label the image of figure ABCD under the following sequence of transformations. •

Dilation centered at the origin with scale factor  _1

•

Reflection across the y-axis

y 10 9 8 7 6 5 4 3 2 1 −10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10

1 2 3 4 5 6 7 8 9 10

For problems 12–14, combine like terms. 12. (5y + 11) − (3y + 1)

13. 7(4a − 3a + 9)

14. 3c + 9 − 4(c + 1)

252

P r act i c e

EM2_0803SE_D_L15_daily_practice.indd 252

10/19/2020 1:00:03 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 16

LESSON Name

Date

Similar Right Triangles Comparing Ratios 1. Consider △ ABC.

B 4 A

a. Draw the image of △ ABC under a dilation with center A and scale factor 2. Label the image △ ADE and label any known side lengths. b. Draw the image of △ ABC under a dilation with center A and scale factor 3. Label the image △ AFG and label any known side lengths.

c. Find the value of the ratio of the height to the base of each triangle. What do you notice?

EM2_0803SE_D_L16_classwork.indd 253

253

10/10/2020 1:26:04 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 16

Finding Unknown Lengths

¯? 2. Given △WIN ∼ △MAT, what is the length of  WI 

N 8 I

A 5 T

254

L esson

EM2_0803SE_D_L16_classwork.indd 254

10/10/2020 1:26:05 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 16

¯? 3. Given △MUG ∼ △JAR, what is the length of  JA 

M 9.6 6 U

EM2_0803SE_D_L16_classwork.indd 255

4.8

L esson

255

10/10/2020 1:26:05 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 16

Using the Pythagorean Theorem

¯? 4. Given △NET ∼ △HAD, what is the length of  DH 

3 T

7.2

N D

256

L esson

EM2_0803SE_D_L16_classwork.indd 256

2.5

10/10/2020 1:26:05 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 16

EXIT TICKET Name

Date

In the diagram, △PQR ∼ △ZYX.

P Z a

7.5

3.62 X Q

5.43

Jonas and Sara each write a proportion to solve for the unknown side length a. Jonas

Sara

3.62 a ___   = ____

7.5 a ____   = ____  

7.5

5.43

3.62

5.43

a. Whose proportion is correct? Explain.

¯ is about b. Jonas and Sara use the Pythagorean theorem to determine that the length of  PR  9.26. Write a proportion to solve for c.

c. What is another way to solve for c?

EM2_0803SE_D_L16_exit_ticket.indd 257

257

10/10/2020 1:25:53 PM

EM2_0803SE_D_L16_exit_ticket.indd 258

10/10/2020 1:25:53 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 16

RECAP Name

Date

Similar Right Triangles In this lesson, we •

applied dilations to create similar right triangles.

•

used within-triangle ratios and between-triangle ratios to find unknown side lengths in similar right triangles.

Examples

1. Given △PQR ∼ △ ABC, what is the length of AC   ¯ ?

7.5 3 P

AC ___ PR ___ BC  = QR    

This proportion uses

AC 3_ ___ 7.5  = 5 

within-triangle ratios.

7.5(___ 7.5 ) = 7.5(5_  ) AC

AC = 4.5

C AC ___ BC ___ PR  =  QR 

___   = ___   5 3 AC

7.5

3(___ 3 ) = 3(___ 5 ) AC

A proportion that uses between-triangle ratios can also be written.

7.5

AC = 4.5

¯ is 4.5 units. The length of  AC 

EM2_0803SE_D_L16_lesson_recap.indd 259

259

10/10/2020 6:21:41 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 16

¯, and   TU  ¯,   ST  ¯. 2. Given △DEF ∼ △STU, find the lengths of  EF 

E U

2.5 F

¯. Let c represent the length of  EF 

52 + 122 = c2 25 + 144 = c2 169 = c2

Use within-triangle ratios or between-triangle ratios to find the length of ST    ¯ or TU    ¯.

13 = c ¯ is 13 units. The length of  EF  ST SU ___     = ___     DE DF ST 2.5 __   = ___  12

12(__ 12 ) = 12(___ 5 ) ST

2.5

ST = 6 ¯ is 6 units. The length of  ST  TU ___ EF ___ SU  =  DF 

= __ 5  ___ 2.5 TU

2.5(___ 2.5 ) = 2.5(__ 5 ) TU

TU = 6.5

Use the Pythagorean theorem, within-triangle ratios, or between-triangle ratios to find the length of the remaining side.

¯ is 6.5 units. The length of  TU 

260

R ecap

EM2_0803SE_D_L16_lesson_recap.indd 260

10/10/2020 6:21:41 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 16

PRACTICE Name

Date

1. △ ABC is a right triangle with side lengths 7.5, 10, and 12.5. Which sets of side lengths form triangles similar to △ ABC? Choose all that apply.

A. 1.5, 2, 2.5 B. 3, 4, 5 C. 6, 8, 9.5

D. 10, 12.5, 15 E. 15, 20, 25 2. △JKL and △MNP are similar triangles.

a. Write a proportion by using within-triangle ratios.

b. Write a proportion by using between-triangle ratios.

EM2_0803SE_D_L16_daily_practice.indd 261

261

10/10/2020 1:28:18 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 16

3. Given △GHI ∼ △WVU, what is the length of GI   ¯ ?

14.4

W 6 4.8 U

4. Given △DEF ∼ △UVW, what is the length of UW   ¯ ?

262

P ractice

EM2_0803SE_D_L16_daily_practice.indd 262

10/10/2020 1:28:18 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 16

¯,   JL  ¯, and   RT  ¯. 5. Given △JKL ∼ △RST, find the lengths of  ST 

K 7.5 J

¯,   PQ  ¯, and   QR  ¯. 6. Given △MNO ∼ △PQR, find the lengths of  NO 

9 M 82

EM2_0803SE_D_L16_daily_practice.indd 263

P ractice

263

10/10/2020 1:28:18 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 16

7. In the diagram, △JKL ∼ △MNP.

K b 6.5 c J 6 L P

a. Maya and Dylan each write a proportion to solve for the unknown side length a. Maya

Dylan

__ ___   a   = 6.5  

___ __   a   = 24 6  6.5

Whose proportion is correct? Explain.

b. Maya and Dylan each write an equation to solve for the unknown side length c. Maya

Dylan

6 2 + c 2 = 6.5 2

__   c   = ___   6   24

6.5

Whose equation is correct? Explain.

264

P ractice

EM2_0803SE_D_L16_daily_practice.indd 264

10/10/2020 1:28:18 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 16

8. Given △STU ∼ △VTW, solve for x.

T x

6 7.5

22.5 U

Remember For problems 9 and 10, solve for x. 9. 2x + 4 + 3x = 14

EM2_0803SE_D_L16_daily_practice.indd 265

10. 3x − 7 − 5x = −16

P ractice

265

10/10/2020 1:28:19 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 16

11. Determine whether △ ABC is similar to △JKL. Explain how you know.

72°

43°

72° C

43°

12. Which products are equivalent to 1? Choose all that apply. 1  ) A. 3(−_ 3

B. 0.25 (4) 1 C. −_   (−7) 7 2 D. −_   (−2.5) 5 E. −1(−0.1)

266

P ractice

EM2_0803SE_D_L16_daily_practice.indd 266

10/10/2020 1:28:19 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 17

LESSON Name

Date

Similar Triangles on a Line Comparing Right Triangles

Finding Unknown Side Lengths

EM2_0803SE_D_L17_classwork.indd 267

267

10/10/2020 1:27:57 PM

EM2_0803SE_D_L17_classwork.indd 268

10/10/2020 1:27:57 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 17

EXIT TICKET Name

Date

In the diagram, the right triangles have horizontal and vertical legs, and the hypotenuses lie on the same line.

D E

DF . Explain. Determine the value of  ___ EF

EM2_0803SE_D_L17_exit_ticket.indd 269

269

10/10/2020 1:27:35 PM

EM2_0803SE_D_L17_exit_ticket.indd 270

10/10/2020 1:27:35 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 17

RECAP Name

Date

Similar Triangles on a Line In this lesson, we •

determined that right triangles with horizontal and vertical legs and with hypotenuses that lie on the same line are similar triangles.

Examples For problems 1 and 2, the right triangles have horizontal and vertical legs, and the hypotenuses of the two triangles lie on the same line. Find the value of __  a  . b

9 D

Because the triangles are similar, use within-triangle ratios to find the value of _  a  . b

F R

Right triangles with a horizontal leg, a vertical leg, and hypotenuses that lie on the same line are similar triangles.

a S

The value of __  a  is  _7  . 3

The values of the within-triangle ratios are equivalent.

EM2_0803SE_D_L17_lesson_recap.indd 271

T 1__   a ST 2   =   =     = 1   2 4 b FS

_ ___ __ _

1 2

a EF 21 7   =     =   =   3 9 b DE b

_ __ _ _ The value of __  a  is  _1  . b

 __ 1 

__  2  can be written 2 as _  1  ÷ 2 , or _  1  ⋅ _  1  . 2

2 2

271

11/18/2020 11:33:02 AM

EM2_0803SE_D_L17_lesson_recap.indd 272

10/10/2020 1:27:26 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 17

PRACTICE Name

Date

For problems 1–6, the right triangles have horizontal and vertical legs, and the hypotenuses of the two triangles lie on the same line. Find the value of __  a . b

9 39

b a

12 3

6.75

6.75 b

EM2_0803SE_D_L17_daily_practice.indd 273

273

10/10/2020 1:27:46 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 17

b a

a 12

12 6

¯ and   VW  ¯ are vertical, and ___ 7. In the diagram,  EF    EF = _  1 . What are possible values for the lengths DE 3 ¯ and   UV  ¯? Choose all that apply. of   VW 

D U

V W F

A. VW = 0.5, UV = 1.5 1 B. VW = _   , UV = _1 2 6

C. VW = 6, UV = 2

1 D. VW = _ , UV = 1 3

E. VW = 2.1, UV = 0.7

274

P r act i c e

EM2_0803SE_D_L17_daily_practice.indd 274

10/31/2020 3:18:53 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 17

¯ and   XZ  ¯ and   XY  ¯ are vertical, and  AC  ¯ lie on the same line. What are possible 8. In the diagram,  AB  ¯ and   YZ  ¯ ? Choose all that apply. The diagram is not drawn to scale. values for the lengths of  XY 

B 3 C 6 A

A. XY = 15, YZ = 7.5 B. XY = 12.5, YZ = 25 C. XY = 18.4 , YZ = 9.2 D. XY = 14, YZ = 7 E. XY = 21.3, YZ = 42.6

EM2_0803SE_D_L17_daily_practice.indd 275

P ractice

275

10/10/2020 1:27:47 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 17

9. Maya states that △RST ∼ △GHI because they are both right triangles that lie on the same line. Do you agree with Maya? Explain.

𝓁 T S I

G R

¯ is vertical. ¯ is horizontal, and  BC  10. In the diagram, the hypotenuse of △ ABC lies on the line,  AB  Draw and label a second right triangle △ AGH with the following conditions: •

• • •

Point G is halfway between points of A and B. ¯. GH    ¯ forms a right angle with  AB  ¯. Point H is on  AC 

¯ and   HG  ¯. Include side lengths  AG 

C 32.5 12.5 A

276

P ractice

EM2_0803SE_D_L17_daily_practice.indd 276

10/10/2020 1:27:47 PM

EUREKA MATH2 8 ▸ M3 ▸ TD ▸ Lesson 17

¯, and the hypotenuse lies ¯, a horizontal leg,  BC  11. In the diagram, △ ABC has a vertical leg,  AB  ⟷ on  AC . Draw and label a second right triangle with the following conditions: •

The triangle shares either ∠ A or ∠C. ¯. The triangle has a hypotenuse on  AC 

•

The drawing of the triangle includes all three side lengths rounded to the tenths place.

•

B 1

Remember For problems 12 and 13, solve for x. 1 x + 6 − x = − 5 12. _ 2

EM2_0803SE_D_L17_daily_practice.indd 277

1 13. _ x − 1 + 1_ x = 9 2 4

P ractice

277

10/10/2020 1:27:47 PM

EUREKA MATH2

8 ▸ M3 ▸ TD ▸ Lesson 17

14. Is △ ABE similar to △CDE ? Explain.

B 47°

22°

111°

15. Identify the relationship as either proportional or not proportional. Explain.

Total Votes Collected at Local Precinct

y 160 140 120 100 80 60 40 20

Number of Hours After Opening

278

P ractice

EM2_0803SE_D_L17_daily_practice.indd 278

10/10/2020 1:27:48 PM

EUREKA MATH2 8 ▸ M3

Mixed Practice

Name

Date

1. Nora makes muffins for the school bake sale. The equation m = 12t represents the total number of muffins m Nora makes with t muffin trays. What is the constant of proportionality?

2. Henry plants a tree and measures its height each month. The table shows his results. Number of Months

Tree Height (inches)

Is the relationship between the number of months and the tree height proportional? Explain.

EM2_0803SE_mixed_practice1.indd 279

279

10/10/2020 12:59:51 PM

EUREKA MATH2

8 ▸ M3 ▸ Mixed Practice 1

3. Eve buys sandwiches for a party. The table shows the total cost to buy a different number of sandwiches. Number of Sandwiches

Total Cost (dollars)

47.00

70.50

94.00

117.50

141.00

Is the relationship between the number of sandwiches and the total cost proportional? Explain.

4. The distance from the sun to Earth is about 150,000,000 km. The distance from the sun to Neptune is about 4,500,000,000 km. The distance from the sun to Neptune is how many times as much as the distance from the sun to Earth? Write your answer in scientific notation and in standard form.

280

EM2_0803SE_mixed_practice1.indd 280

10/10/2020 12:59:52 PM

EUREKA MATH2 8 ▸ M3 ▸ Mixed Practice 1

5. Find the value of x.

A 68° x° C 76° B

⟷

6. What is the image of  AB  under a translation 3 units left and 2 units down? y 10 9 8 7 6 5 4 3 2 1 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

9 10

−2 −3 −4 −5 −6 −7 −8 −9 −10

⟷

A. A line segment parallel to  AB 

⟷

B. A line segment that intersects  AB 

⟷

C. A line parallel to  AB 

⟷

D. A line that intersects  AB 

EM2_0803SE_mixed_practice1.indd 281

281

10/10/2020 12:59:52 PM

EUREKA MATH2

8 ▸ M3 ▸ Mixed Practice 1

7. Graph and label the image of figure ABCDEFG under the following sequence of rigid motions. •

Reflection across the y-axis

•

Translation 2 units left and 8 units down y 10 9 8

F G

A B

7 6

5 4 3 2

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

9 10

−2 −3 −4 −5 −6 −7 −8 −9 −10

8. Is △ ABC congruent to △DEF ? Explain.

A E

282

EM2_0803SE_mixed_practice1.indd 282

10/10/2020 12:59:52 PM

EUREKA MATH2 8 ▸ M3

Mixed Practice

Name

Date

1. Indicate whether each number is rational or irrational. Number

Rational

Irrational

0.08333333… 0.101101110… 6.03125 2.44948949847… π 8_ 7

2. Logan and Shawn are trying to decide whether 2.01388888… is a rational or irrational number. •

Logan says the number is rational because the 8 repeats.

•

Shawn says the number is irrational because the 0, 1, and 3 appear only once and do not repeat.

Who is correct? Explain.

EM2_0803SE_mixed_practice2.indd 283

283

10/10/2020 1:04:12 PM

EUREKA MATH2

8 ▸ M3 ▸ Mixed Practice 2

For problems 3–6, fill in the blank with an approximation of the number rounded to the nearest tenth. Then plot your approximation on the number line. — 5  ≈ 3. √

–5

–4

–3

–2

–1

–5

–4

–3

–2

–1

–5

–4

–3

–2

–1

–5

–4

–3

–2

–1

4. −π ≈

— 5. √17  ≈

— 6. −√2  ≈

For problems 7–9, compare the numbers by using the < or > symbol. — −√7  7. −π

— 8. 2 · √ 2  9. π + 3

284

EM2_0803SE_mixed_practice2.indd 284

— 2·√ 3  —

√ 50 

10/10/2020 1:04:12 PM

EUREKA MATH2 8 ▸ M3 ▸ Mixed Practice 2

⟶

10. A translation along  AB  maps figure M onto which figure? A. Figure A B. Figure B C. Figure C D. Figure D

B C D

EM2_0803SE_mixed_practice2.indd 285

285

10/10/2020 1:04:13 PM

EUREKA MATH2

8 ▸ M3 ▸ Mixed Practice 2

11. Consider △ ABC shown in the coordinate plane. y

10 9

8 7 6

5 4 3

2 1 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

9 10

−2 −3 −4 −5 −6 −7 −8 −9 −10

a. Graph the image of △ ABC under the following rigid motions. •

Reflection across the y-axis

•

Reflection across the x-axis

Label the image △DEF.

b. Graph the image △ ABC under a 180° rotation around the origin. Label the image △GHI. c. Are △DEF and △GHI in the same location? Explain with the coordinates of the vertices.

286

EM2_0803SE_mixed_practice2.indd 286

11/19/2020 11:14:56 AM

EUREKA MATH2 8 ▸ M3 ▸ Mixed Practice 2

12. Let △ ABC be a right triangle with an area of 30 square units. The triangle is located in the first quadrant. One side of △ ABC is shown. y 10 9

8 7 6 5 4 3

2 1 0

a. Determine one possible location for point C. Draw △ ABC.

b. What is the perimeter of △ ABC?

EM2_0803SE_mixed_practice2.indd 287

287

10/30/2020 10:46:35 AM

EM2_0803SE_mixed_practice2.indd 288

10/10/2020 1:04:13 PM

EUREKA MATH2 8 ▸ M3 ▸ TC ▸ Lesson 10 ▸ Transformations

Apply the given transformation. 1. Graph the image of figure M under a reflection across the y-axis. Label the image R. 2. Graph the image of figure M under a translation 9 units left and 5 units down. Label the image V. 3. Graph the image of figure M under a dilation centered at the origin with scale factor 2. Label the image D.

y 10 9 8 7 6 5 4 3

2 1 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −1

−2 −3 −4 −5 −6 −7 −8 −9 −10

EM2_0803SE_C_L10_template_transformation.indd 289

289

10/10/2020 1:07:45 PM

EM2_0803SE_C_L10_template_transformation.indd 290

2/24/2021 6:26:15 PM

EUREKA MATH2 8 ▸ M3 ▸ Sprint ▸ Apply Properties of Exponents for a Power Raised to a Power

Sprint Apply the properties and definitions of exponents to write an equivalent expression with positive exponents. Assume x and y are nonzero. 1.

(76) 3

(x   2 y  3)5

EM2_0803SE_sprint_apply_properties_of_exponents_power_of_a_power.indd 291

291

12/3/2020 3:21:34 PM

EUREKA MATH2

8 ▸ M3 ▸ Sprint ▸ Apply Properties of Exponents for a Power Raised to a Power

Number Correct:

Apply the properties and definitions of exponents to write an equivalent expression with positive exponents. Assume x and y are nonzero. 1.

(53 )2

23.

(3x)  8

( 53 ) 3

24.

( 3x)  9

( 53 ) 4

25.

( 220) 50

( 53 ) 5

26.

( 220) 55

(x   5)3

27.

(220)60

(x   6)3

28.

(2270)0

(x   7)3

29.

(x y  3)10

(x  8)3

30.

(x   4 y)  9

(x   3 y  6)2

31.

(x   3 y)  8

10.

(x   3 y  6)3

32.

(x   0 y)  7

11.

(x   3 y  6)4

33.

12.

(x   3 y  6)5

34.

13.

(22 x  2)2

35.

14.

( 23 x  4) 2

36.

15.

(24 x  6)2

37.

16.

( 25 x  8) 2

38.

17.

( 4 x  2) 3

39.

18.

(6 x  4)4

40.

19.

(8 x  4)5

41.

20.

(10 x  5)6

42.

21.

( 3 x)  5

43.

22.

(3 x)  7

44.

292

EM2_0803SE_sprint_apply_properties_of_exponents_power_of_a_power.indd 292

2 4  x __   

( y   3)

4 4 x__   

( y  6)

6 4 x__   

( y  8)

x   8  4   ___

(y  10)

((1_  ) )

4 5

2 5 10 1 ((  ) ) 2 6 20 ((1  ) ) 2 7 30 ((1  ) ) 2 5 2 ((2  ) ) 3 10 3 ((2  ) ) 3 20 4 2 ((  ) ) 3 30 5 ((2  ) ) 3

10/30/2020 6:28:29 AM

EM2_0803SE_sprint_apply_properties_of_exponents_power_of_a_power.indd 293

10/19/2020 2:44:49 PM

EUREKA MATH2

8 ▸ M3 ▸ Sprint ▸ Apply Properties of Exponents for a Power Raised to a Power

Number Correct: Improvement:

Apply the properties and definitions of exponents to write an equivalent expression with positive exponents. Assume x and y are nonzero. 1.

(54) 2

23.

(3x)  9

( 55) 3

24.

( 3x)  10

( 56) 4

25.

( 220) 30

( 57 ) 5

26.

( 220) 35

(x   4) 3

27.

(220)40

(x   5) 3

28.

(2250)0

(x   6) 3

29.

(x y  3)5

(x   7) 3

30.

(x   4 y)  4

(x   2 y  5) 2

31.

(x   3 y)  3

10.

(x   2 y  5) 3

32.

(x   0 y)  2

11.

(x   2 y  5) 4

33.

12.

(x   2 y  5) 5

34.

13.

(22 x  2) 3

35.

14.

( 23 x  4) 3

36.

15.

(24 x  6)3

37.

16.

( 25 x  8) 3

38.

17.

( 4 x  3) 4

39.

18.

(6 x  5) 5

40.

19.

(8 x  5) 6

41.

20.

(10x  6) 7

42.

21.

( 3x)  6

43.

22.

(3x)  8

44.

294

EM2_0803SE_sprint_apply_properties_of_exponents_power_of_a_power.indd 294

2 5 x__   

( y   3)

  4  5 x__

( y  6)

6 5 x__   

( y  8)

x   8   5   ___

( y  10)

((1_  ) ) 2

10 3

((1_  ) ) 2

20 4

((1_  ) ) 2

30 5

((1_  ) ) 2

40 6

((2_  ) ) 3

6 5

((_2  ) ) 3

7 10

((2_  ) ) 3

8 30

( (2_  ) ) 3

9 40

10/19/2020 2:44:49 PM

EUREKA MATH2 8 ▸ M3 ▸ Sprint ▸ Compare Numbers and Square Roots

Sprint Use the symbols >, =, or < to make a true statement. 1. 2.

—

√ 9 

— √9 

2 3

EM2_0803SE_sprint_compare_numbers_and_square_roots_using _symbols.indd 295

295

12/3/2020 3:23:33 PM

EUREKA MATH2

8 ▸ M3 ▸ Sprint ▸ Compare Numbers and Square Roots

Number Correct:

Use the symbols >, =, or < to make a true statement. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

—

√ 1 

—

23.

— √1 

24.

—

25.

— √4 

26.

—

27.

— √4 

28.

—

29.

—

30.

— √16 

31.

—

32.

— √64 

33.

—

34.

— √81 

35.

—

36.

— √100 

37.

—

38.

—

39.

— √120 

40.

—

41.

— √121 

42.

—

43.

44.

√ 1  √ 1 

√ 4  √ 4 

√ 16  √ 16 

√ 65  √ 80 

√ 99 

√ 101 

√ 120  √ 122 

296

EM2_0803SE_sprint_compare_numbers_and_square_roots_using _symbols.indd 296

—

√ 9 

2.9

—

√ 9 

3.1

— √ 9 

—

3.14

—

√ 9 

— √0.16 

— √16 

0.4

√ 0.16 

0.8

—

√ 0.64 

—

√ 0.64 

0.8

— √0.64 

0.08

—

√ 0.64 

0.008

— — √100  + √ 81  —

—

√ 100  + √ 81 

— — √100  + √ 81  —

√ 100  + √ 81 

— — √100  + √ 81 

—

√ 100  − √ 81 

—

— — √100  − √ 81 

√ 100  − √ 81 

— — √100  − √ 81 

—

√ 100  − √ 81 

—

√ 19 

√ 100  − √ 81 

—

√ 9 

9 —

√ 1 

—

√ 9 

10/10/2020 1:09:37 PM

EM2_0803SE_sprint_compare_numbers_and_square_roots_using _symbols.indd 297

10/10/2020 1:09:37 PM

EUREKA MATH2

8 ▸ M3 ▸ Sprint ▸ Compare Numbers and Square Roots

Number Correct: Improvement:

Use the symbols >, =, or < to make a true statement. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

—

√ 1 

—

23.

— √1 

24.

—

25.

— √9 

26.

—

27.

— √9 

28.

—

29.

—

30.

— √25 

31.

—

32.

— √36 

33.

—

34.

— √49 

35.

—

36.

— √80 

37.

—

38.

—

39.

— √100 

40.

—

41.

— √120 

42.

43.

44.

√ 1  √ 1 

√ 9  √ 9 

√ 25  √ 25 

√ 37 

√ 48  √ 81  √ 82 

√ 100 

—

√ 125 

298

EM2_0803SE_sprint_compare_numbers_and_square_roots_using _symbols.indd 298

—

√ 4 

—

1.9

—

2.01

√ 4 

2.1

— √ 4  √ 4 

2.11

— √0.25 

—

√ 0.25 

0.5

— √25 

0.1

—

√ 25 

—

√ 0.49 

0.7

— √0.49 

0.07

—

√ 0.49 

0.007

— — √100  + √ 25 

—

√ 100  + √ 25 

— — √100  + √ 25  —

—

√ 100  + √ 25 

— — √100  + √ 25 

—

√ 100  − √ 25 

—

10 15

—

√ 25 

—

√ 75 

—

√ 100  − √ 25 

— — √100  − √ 25  —

√ 100  − √ 25 

— — √100  − √ 25 

—

√ 100  − √ 25 

—

√ 4 

—

√ 25 

10/10/2020 1:09:38 PM

EUREKA MATH2 8 ▸ M3 ▸ Sprint ▸ One-Step Equations–Multiplication

Sprint Find the value of p in each equation. 1.

2p = 8

3p = 24

EM2_0803SE_sprint_one_step_equations_multiplication_positive_only.indd 299

299

12/3/2020 3:25:11 PM

EUREKA MATH2

8 ▸ M3 ▸ Sprint ▸ One-Step Equations–Multiplication

Number Correct:

Find the value of p in each equation.

2p = 2

19.

1   p = 4 2

2p = 4

20.

p   = 4 2

2p = 6

21.

1   p = 6 2

2p = 10

22.

1   p = 2 3

2p = 20

23.

p   = 2 3

3p = 12

24.

p   = 4 3

3p = 18

25.

1   p = 6 3

3p = 21

26.

p   = 6 3

3p = 27

27.

1   p = 8 3

10.

4p = 28

28.

p   = 8 3

11.

4p = 32

29.

2p = _  

12.

5p = 20

30.

2p = _  

13.

5p = 30

31.

3p = _  

14.

5p = 45

32.

3p = _  

15.

6p = 24

33.

3p = _  

16.

6p = 30

34.

3p = _  

17.

6p = 42

35.

4p = _  

18.

8p = 56

36.

5p = _  

300

EM2_0803SE_sprint_one_step_equations_multiplication_positive_only.indd 300

_ _ _

_ _

1 2 1 4 1 2 1 5 1 6 1 9 1 6 1 9

10/10/2020 1:09:17 PM

EM2_0803SE_sprint_one_step_equations_multiplication_positive_only.indd 301

10/10/2020 1:09:17 PM

EUREKA MATH2

8 ▸ M3 ▸ Sprint ▸ One-Step Equations–Multiplication

Number Correct: Improvement:

Find the value of p in each equation.

3p = 3

19.

1   p = 3 2

3p = 6

20.

p   = 3 2

3p = 9

21.

1   p = 5 2

3p = 15

22.

1   p = 3 3

3p = 30

23.

p   = 3 3

4p = 12

24.

p   = 5 3

4p = 20

25.

1   p = 7 3

4p = 24

26.

p   = 7 3

4p = 36

27.

1   p = 9 3

10.

5p = 25

28.

p   = 9 3

11.

5p = 40

29.

3p = _  

12.

6p = 18

30.

3p = _  

13.

6p = 36

31.

4p = _  

14.

6p = 48

32.

4p = _  

15.

8p = 24

33.

4p = _  

16.

8p = 40

34.

4p = _  

17.

8p = 64

35.

5p = _  

18.

9p = 72

36.

6p = _  

302

EM2_0803SE_sprint_one_step_equations_multiplication_positive_only.indd 302

_ _ _

_ _

1 2 1 4 1 2 1 3 1 5 1 7 1 8 1 9

10/10/2020 1:09:17 PM

EUREKA MATH2 8 ▸ M3 ▸ Sprint ▸ Solve Simple Proportions

Sprint Solve each equation.

_ __

x 1   =     6 18

1 5   = x  6

_ _

EM2_0803SE_sprint_solve_simple_proportions.indd 303

303

12/3/2020 3:17:08 PM

EUREKA MATH2

8 ▸ M3 ▸ Sprint ▸ Solve Simple Proportions

Number Correct:

Solve each equation.

_ _

1 x   =     2 4

23.

2 x   =     3 6

1 x   =     2 6

24.

2 x   =     3 9

1 x   =     2 8

25.

x 2   =     3 12

x 1   =     2 10

26.

x 2   =     3 15

x 1   =     5 10

27.

x 2   =     3 18

x 1   =     5 15

28.

x 2   =     3 21

x 1   =     5 20

29.

2 4   = x  3

x 1   =     5 25

30.

2 6   = x  3

x 1   =     8 48

31.

2 8   = x  3

10.

x 1   =     8 64

32.

2 10   = x  3

11.

x 1   =     8 80

33.

2 12   = x  3

12.

1 2   =   2 x

34.

5 25   = x  3

13.

1 3   =   2 x

35.

5 x   =     3 27

14.

1 4   =   2 x

36.

5 55   = x  3

15.

1 5   =   2 x

37.

3 45   = x  4

16.

1 2   =   7 x

38.

3 x   =     4 60

17.

1 3   =   7 x

39.

3 x   =     4 72

18.

1 4   =   7 x

40.

1 x   =     2 5

19.

1 5   =   7 x

41.

1 x   =     2 7

20.

5 1     =   10 x

42.

1 x   =     2 9

21.

6 1     =   10 x

43.

3 x   =     4 14

22.

7 1     =   10 x

44.

3 x   =     4 25

_ _ _ _ _ __ _ __ _ __ _ __ _ __ _ __ _ __ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

__ _ __ _ __ _

304

EM2_0803SE_sprint_solve_simple_proportions.indd 304

_ _ _ __ _ __ _ __ _ __ _ _ _ _ _ _ _ __

_ __ _ __ _ __

_ __ _ __

_ __ _ __ _ _ _ _ _ _ _ __ _ __

10/10/2020 1:08:58 PM

EM2_0803SE_sprint_solve_simple_proportions.indd 305

10/10/2020 1:08:58 PM

EUREKA MATH2

8 ▸ M3 ▸ Sprint ▸ Solve Simple Proportions

Number Correct: Improvement:

Solve each equation.

_ _

1 x =   3 6

23.

3 x =   2 4

1 x =   3 9

24.

3 x =   2 6

x 1 =   3 12

25.

3 x =   2 8

x 1 =   3 15

26.

3 x =   2 10

x 1 =   4 16

27.

3 x =   2 12

x 1 =   4 20

28.

3 x =   2 14

x 1 =   4 24

29.

3 6 = x 2

x 1 =   4 28

30.

3 9 = x 2

31.

3 12 = x 2

32.

3 15 = x 2

100

33.

3 18 = x 2 3 15 = x 5

9. 10.

_ _ _ __ _ __ _ __ _ __ _ __ _ __ x 1 __   = __   x 1 __   = __  

_ _ _ _ _ __ _ __ _ __ _ _ _ _ _ __ _ __

11.

x 1 __   = ___  

12.

1 2 = x 3

34.

13.

1 3 = x 3

35.

3 x =   5 30

14.

1 4 = x 3

36.

3 21 = x 5

15.

1 5 = x 3

37.

4 32 = x 3

16.

1 2 = x 6

38.

x _4 = __   3

17.

1 3 = x 6

39.

x _4 = __   3

18.

1 4 = x 6

40.

x _1 = __   2

19.

1 5 = x 6

41.

x _1 = __   2

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

_ __ _ __ _ __

_ __ _ __

20.

5 1 __   = _ 20

x _1 = __   2

21.

6 1 __   = _

42.

43.

x _3 = __   4

22.

44.

7 1 __   = _

306

EM2_0803SE_sprint_solve_simple_proportions.indd 306

x _3 = __  

10/10/2020 1:08:59 PM

EUREKA MATH2 8 ▸ M3 ▸ Sprint ▸ Solve Two-Step Equations

Sprint Solve each equation. 1.

3x + 1 = 10

3x − 4 = 8

EM2_0803SE_sprint_solve_two_step_equations.indd 307

307

12/3/2020 3:03:23 PM

EUREKA MATH2

8 ▸ M3 ▸ Sprint ▸ Solve Two-Step Equations

Number Correct:

Solve each equation. 1.

2x + 4 = 14

19.

12 = 3(x + 2)

2x + 5 = 15

20.

3x + 6 = 12

2x + 6 = 16

21.

3x + 12 = 6

2x + 7 = 17

22.

−6 = 3x + 12

2x + 8 = 18

23.

3x − 12 = −6

2x − 2 = 6

24.

−3x − 6 = −12

2x − 4 = 6

25.

−3(x − 2) = −12

2x − 6 = 6

26.

6 = −3(x − 2)

2x − 8 = 6

27.

3(x + 2) = −6

10.

2x − 6 = 14

28.

_1   (x + 6) = 3

11.

2x − 6 = 12

29.

3 = _ x + 2

12.

2x + 4 = 18

30.

1 x − 2 = −3 3

13.

2x + 4 = 16

31.

1 x − 3 = 2 3

14.

2(x + 2) = 16

32.

− _ x + 3 = 2

15.

2(x + 2) = 18

33.

−2 = − _ (x − 9)

16.

2(x + 2) = 20

34.

− _ (x − 9) = 3

17.

2(x − 2) = 20

35.

3 = − _ x + 3

18.

2(x − 2) = 18

36.

− _ x + 3 = −3

308

EM2_0803SE_sprint_solve_two_step_equations.indd 308

1 3

10/10/2020 1:08:37 PM

EM2_0803SE_sprint_solve_two_step_equations.indd 309

10/10/2020 1:08:37 PM

EUREKA MATH2

8 ▸ M3 ▸ Sprint ▸ Solve Two-Step Equations

Number Correct: Improvement:

Solve each equation. 1.

3x + 4 = 10

19.

20 = 4(x + 3)

3x + 5 = 11

20.

4x + 12 = 20

3x + 6 = 12

21.

4x + 20 = 12

3x + 7 = 13

22.

−12 = 4x + 20

3x + 8 = 14

23.

4x − 20 = −12

3x − 3 = 6

24.

−4x − 12 = −20

3x − 6 = 6

25.

−4(x − 3) = −20

3x − 9 = 6

26.

12 = −4(x − 3)

3x − 12 = 6

27.

4(x + 3) = −12

10.

3x − 6 = 18

28.

1 (x + 8) = 4 4

11.

3x − 6 = 15

29.

4 = _ x + 2

12.

3x + 9 = 21

30.

1 x − 2 = −4 4

13.

3x + 9 = 18

31.

1 x − 4 = 2 4

14.

3(x + 3) = 18

32.

− _ x + 4 = 2

15.

3(x + 3) = 21

33.

−2 = − _ (x − 16)

16.

3(x + 3) = 24

34.

− _ (x − 16) = 4

17.

3(x − 3) = 24

35.

4 = − _ x + 4

18.

3(x − 3) = 21

36.

− _ x + 4 = −4

310

EM2_0803SE_sprint_solve_two_step_equations.indd 310

1 4

10/10/2020 1:08:37 PM

EUREKA MATH2 8 ▸ M3

Credits Great Minds® has made every effort to obtain permission for the reprinting of all copyrighted material. If any owner of copyrighted material is not acknowledged herein, please contact Great Minds for proper acknowledgment in all future editions and reprints of this module. Cover, Al Held (1928–2005), Pan North IV, 1985, acrylic on canvas, 72 x 84 in., private collection. © 2020 Al Held Foundation, Inc./Licensed by Artists Rights Society (ARS), New York; pages 191, 193 (details), Wassily Kandinsky (1866–1944), Composition 8, July 1923. Oil on canvas. 55 1/4 x 79 inches (140.3 x 200.7 cm). Solomon R. Guggenheim Founding Collection, by gift. The Solomon R. Guggenheim Museum, New York, NY, USA Photo Credit: The Solomon R. Guggenheim Foundation/Art Resource, NY. © 2020 Artists Rights Society (ARS), New York; page 237 (top left), The Papyrus of Ani, papyrus manuscript written in cursive hieroglyphs and illustrated with color miniatures, 19th dynasty of the New Kingdom of ancient Egypt. Photo credit: Science History Images/Alamy Stock Photo, (top right), Robert Duncanson, Blue Hole, Little Miami River. Photo credit: Historic Images/Alamy Stock Photo, (bottom left), Gustav Klimt, The Kiss, 1907–1908. Oil on canvas. Austrian Gallery Belvedere, Vienna, Austria. Photo credit: PAINTING/Alamy Stock Photo, (bottom right) Aztec history murals by Diego Rivera in the National Palace, Palacio Nacional, Mexico City, Mexico, © 2021 Banco de México Diego Rivera Frida Kahlo Museums Trust, Mexico, D.F./Artists Rights Society (ARS), New York. Photo credit: dbimages/ Alamy Stock Photo; page 238, (top right), Gustave Caillebotte, Study for Le Pont de L’Europe, 1876, Albright-Knox Art Gallery, Buffalo, New York, USA, North America. Photo credit: Peter Barritt/Alamy Stock Photo, (bottom left), Meindert Hobbema, The Avenue at Middelharnis, 1689, Dutch, The Netherlands. Photo credit: Peter Horree/Alamy Stock Photo; page 247, Anne08/Shutterstock.com; page 250, Rauf Aliyev/Shutterstock.com; All other images are the property of Great Minds. For a complete list of credits, visit http://eurmath.link/media-credits.

EM2_0803SE_Credits.indd 311

311

10/31/2020 11:46:32 AM

8 ▸ M3

EUREKA MATH2

Acknowledgments Adriana Akers, Amanda Aleksiak, Tiah Alphonso, Lisa Babcock, Christopher Barbee, Reshma P Bell, Chris Black, Erik Brandon, Beth Brown, Amanda H. Carter, Leah Childers, David Choukalas, Mary Christensen-Cooper, Cheri DeBusk, Jill Diniz, Mary Drayer, Karen Eckberg, Dane Ehlert, Samantha Falkner, Scott Farrar, Kelli Ferko, Krysta Gibbs, Winnie Gilbert, Danielle Goedel, Julie Grove, Marvin E. Harrell, Stefanie Hassan, Robert Hollister, Rachel Hylton, Travis Jones, Kathy Kehrli, Raena King, Emily Koesters, Liz Krisher, Alonso Llerena, Gabrielle Mathiesen, Maureen McNamara Jones, Pia Mohsen, Bruce Myers, Marya Myers, Kati O’Neill, Ben Orlin, April Picard, John Reynolds, Bonnie Sanders, Aly Schooley, Erika Silva, Hester Sofranko, Bridget Soumeillan, Ashley Spencer, Danielle Stantoznik, Tara Stewart, James Tanton, Cathy Terwilliger, Cody Waters, Valerie Weage, Allison Witcraft, Caroline Yang Trevor Barnes, Brianna Bemel, Adam Cardais, Christina Cooper, Natasha Curtis, Jessica Dahl, Brandon Dawley, Delsena Draper, Sandy Engelman, Tamara Estrada, Soudea Forbes, Jen Forbus, Reba Frederics, Liz Gabbard, Diana Ghazzawi, Lisa Giddens-White, Laurie Gonsoulin, Nathan Hall, Cassie Hart, Marcela Hernandez, Rachel Hirsh, Abbi Hoerst, Libby Howard, Amy Kanjuka, Ashley Kelley, Lisa King, Sarah Kopec, Drew Krepp, Crystal Love, Maya Márquez, Siena Mazero, Cindy Medici, Ivonne Mercado, Sandra Mercado, Brian Methe, Patricia Mickelberry, Mary-Lise Nazaire, Corinne Newbegin, Max Oosterbaan, Tamara Otto, Christine Palmtag, Andy Peterson, Lizette Porras, Karen Rollhauser, Neela Roy, Gina Schenck, Amy Schoon, Aaron Shields, Leigh Sterten, Mary Sudul, Lisa Sweeney, Samuel Weyand, Dave White, Charmaine Whitman, Nicole Williams, Glenda Wisenburn-Burke, Howard Yaffe

312

EM2_0803SE_acknowledgements.indd 312

12/28/2020 2:01:11 PM

Talking Tool Share Your Thinking

I know . . . . I did it this way because . . . . The answer is

because . . . .

My drawing shows . . . . I agree because . . . .

Agree or Disagree

That is true because . . . . I disagree because . . . . That is not true because . . . . Do you agree or disagree with

Ask for Reasoning

? Why?

Why did you . . . ? Can you explain . . . ? What can we do first? How is

Say It Again

related to

I heard you say . . . . said . . . . Another way to say that is . . . . What does that mean?

EM2_0803SE_acknowledgements.indd 313

10/10/2020 1:11:05 PM

Thinking Tool When I solve a problem or work on a task, I ask myself Before

Have I done something like this before? What strategy will I use? Do I need any tools?

During

Is my strategy working? Should I try something else? Does this make sense?

After

What worked well? What will I do differently next time?

At the end of each class, I ask myself

What did I learn? What do I have a question about?

EM2_0803SE_acknowledgements.indd 314

10/10/2020 1:11:06 PM

MATH IS EVERYWHERE Do you want to compare how fast you and your friends can run? Or estimate how many bees are in a hive? Or calculate your batting average? Math lies behind so many of life’s wonders, puzzles, and plans. From ancient times to today, we have used math to construct pyramids, sail the seas, build skyscrapers—and even send spacecraft to Mars. Fueled by your curiosity to understand the world, math will propel you down any path you choose. Ready to get started?

Module 1 Scientific Notation, Exponents, and Irrational Numbers Module 2 Rigid Motions and Congruent Figures Module 3 Dilations and Similar Figures Module 4 Linear Equations in One and Two Variables Module 5 Systems of Linear Equations Module 6 Functions and Bivariate Statistics

What does this painting have to do with math? Abstract expressionist Al Held was an American painter best known for his “hard edge” geometric paintings. His bright palettes and bold forms create a three-dimensional space that appears to have infinite depth. Held, who sometimes found inspiration in architecture, would often play with the viewer’s sense of visual perception. While most of Held’s artworks are paintings, he also worked in mosaic and stained glass. On the cover Pan North IV, 1985 Al Held, American, 1928–2005 Acrylic on canvas Private collection Al Held (1928–2005), Pan North IV, 1985, acrylic on canvas, 72 x 84 in, private collection. © 2020 Al Held Foundation, Inc./Licensed by Artists Rights Society (ARS), New York

ISBN 978-1-64497-133-8

781644 971338