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Use What You Know

Lesson 20 Transformations and Similarity

Transformations and SimilarityIntroductionLesson 20

You learned that if you reflect, translate, or rotate a shape, the figure and its image are congruent. In this lesson you will learn about a transformation that changes the size of a polygon.

Michael is using a photo editing program to adjust the size of photos for a yearbook. To avoid distorting the image, he pulls a corner of the photo along a line, as shown by the dotted line.

What type of transformation transforms nOAB to nOCD?

1

1

Z Dy

A

B

O Cx

Use the math you already know to solve this problem.

a. Find the lengths of segments OA, AB, OC, and CD. What do you notice about the lengths

of corresponding sides?

b. Compare the coordinates of the vertices of nOAB and nOCD. What do you notice about

the coordinates?

c. Remember that the lengths of corresponding sides of similar triangles are in proportion. Are nOAB and nOCD similar triangles? Explain why or why not.

M.8.18

M.8.19

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Find Out More


The transformation that transforms nOAB to nOCD is called a dilation. A dilation is a transformation in which the original figure and the image are similar.

In a dilation, the ratio of the length of a side of the image to the length of the corresponding side of the original figure is called the scale factor. The center of a dilation is the point that is transformed to itself by the dilation.

In the dilation on the previous page, the ratio OC ··· OA is 4 ·· 2 and the ratio DC ··· BA is 6 ·· 3 . So the scale

factor is 2. The center of dilation is the origin, or point O.

In this lesson, you will work with scale factors greater than 0. If the scale factor of the dilation is greater than 1, the image is larger in size than the original figure. If the scale factor is between 0 and 1, the image is smaller in size than the original figure.

Reflect

1 Write the coordinates of the vertices of the image of the parallelogram below after a

dilation with scale factor 3 and center O.

x

y

O

2

4

6

8

10

–10 –8 –6 –4 –2 2 4 6 8 10

–8

–6

–4

–2

–10

Learn About


Modeled and Guided InstructionLesson 20


Combining Dilations and Other Transformations

Read the problem below. Then learn about combining a dilation with another transformation.

In the diagram below, nPQR is similar to nLMN. Describe the sequence of transformations that transforms nPQR to nLMN.

y

x

O L

R

Q

N

M

P

Model It You can describe the transformation with words.

nPQR was flipped, or reflected, over the y-axis.

nPQR was increased in size, or dilated about center O, with a scale factor of 3 ·· 2 .

Model It You can describe the change in the coordinates of the vertices of the triangle.

P(22, 0) was transformed to L(3, 0).

Q(24, 4) was transformed to M(6, 6).

R(0, 2) was transformed to N(0, 3).

Each x-coordinate has opposite signs and was multiplied by 3 ·· 2 .

Each y-coordinate was multiplied by 3 ·· 2 .

©Curriculum Associates, LLC Copying is not permitted. 179Lesson 20 Transformations and Similarity

Connect It Now you will explore how to combine a dilation with another transformation.

2 What line of reflection is used to transform nPQR to nLMN? Explain.

3 What scale factor and center are used to transform the reflection image of nPQR to nLMN?

4 Does it matter if nPQR is first dilated, then reflected, or if it is first reflected, then dilated? Explain why or why not.

5 Suppose nLMN is the original figure and nPQR is the image of the sequence of transformations. Would this change the description of the transformations?

6 Problem 5 shows that a dilation with a scale factor between 0 and 1 shrinks the polygon. What do you think is the effect of a dilation with a scale factor of 1?

Try It

7 UVWO is similar to DEFO. Describe the transformation or sequence of transformations that transforms UVWO to DEFO.

x

y

OU

VWD

1

E

F 2

Guided Practice

Practice


Lesson 20



Study the example below. Then solve problems 8–10.

Example

ΔABC is reflected across the x-axis and dilated with scale factor 1 ·· 2 and

center O. What are the coordinates of the vertices of the image

of ΔABC?

y

x

O

B

A C

y

x

O

B

AC

Solution

8 The coordinates of the vertices of nOPQ are O(0, 0); P(4, 0); and

Q(2, 24). nOPQ is dilated with center O and scale factor 3 ·· 2 . Sketch the

dilation. What quadrant or quadrants contain the image of nOPQ?

Show your work.

Solution

Pair/ShareHow else could you make these transformations?

Pair/ShareDo you think a dilation with center (0, 0) moves a polygon from one quadrant to another quadrant?

Start by graphing nOPQ.

The student reflected the triangle across the x-axis, then multiplied each of the coordinates by 1 ·· 2 .

y

x

O

1

The coordinates of the vertices of the final image are

(1, 0); (3, 0); and (2, 2).


9 The coordinates of the vertices of a trapezoid are (0, 0), (2, 4), (5, 4), and (7, 0). The trapezoid is dilated with scale factor 2 and center (0, 0). What are the coordinates of the vertices of the image of the trapezoid?

Solution

10 What two transformations could transform the smaller square to the larger square?

O

2

x

y6

A Dilation with center O and scale factor 3; rotation 45° about O

B Dilation with center O and scale factor 3; rotation 90° about O

C Dilation with center O and scale factor 1 ·· 3 ; rotation 45° about O

D Dilation with center O and scale factor 1 ·· 3 ; rotation 180° about O

Hattie chose C as the correct answer. Why is her answer incorrect?

Pair/ShareDid you need to make a sketch for this problem?

Pair/ShareWhy is the order in the description of a dilation important?

When the center of dilation is at the origin, what effect does the scale factor have on the coordinates?

Make sure you consider both the scale factor and degree of rotation as you look for the correct answer.

Independent Practice

Practice


Lesson 20



Solve the problems.

1 An equilateral triangle with side length 1.5 is dilated with a scale factor of 4. What is the length of one side of the image of the triangle?

A 0.375

B 3

C 1.5

D 6

2 Triangle ABC is shown on the coordinate plane. Shade in the points that represent the vertices of triangle A’B’C’ after a dilation using a scale factor of 2 with the center of dilation at the origin. Then, connect the vertices A’, B’, C’ to form the new triangle.

3 On the coordinate plane below, triangle ABC was rotated 180 degrees around the origin and then dilated by a scale factor of 2 with the center of dilation at the origin to form the blue triangle, where x, y, and z represent the side lengths of the blue triangle.

Complete the proportion below by entering x, y, or z in the appropriate denominator.

AB 5

BC 5

AC

A

B

Cx

y

O

–6

–6 6

6

–4

–2

–4 –2 2 4

2

4

z

A

BC

y

x

x

y

O

–6

–6 6

6

–4

–2

–4 –2 2 4

2

4

Self Check


Go back and see what you can check off on the Self Check on page 159.

4 Describe at least two different transformations or sequences of transformations that transform square A to square B.

A

B

x

y

O

5 Sketch a trapezoid on a coordinate plane, then choose two dilations with different scale factors. Draw the image of the trapezoid after each dilation. Are the sides of the trapezoid that are parallel in the original figure parallel in each image? Do you think that your result will be true for any dilation? Explain why or why not.

x

y

O

–6

–6 6

6

–4

–2

–4 –2 2 4

2

4

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