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©Curriculum Associates, LLC Copying is not permitted. 211Lesson 20 Transformations and Similarity

Name: Transformations and Similarity

Lesson 20

Prerequisite: Use Transformations to Identify Congruent Figures

Study the example problem showing how to use transformations to identify two congruent figures. Then solve problems 1–6.

1 When Polygon KLMN in the example was translated, how did the angle and line properties change? Explain

2 If you rotate Polygon KLMN 180° about the origin, how would the measures of the angles in the image compare to the measures of the corresponding angles in the original fi gure?

Example

Polygon KLMN is translated 3 units down and 5 units to the left Polygon K9L9M9N9 is its image Are Polygon KLMN and its image congruent?

Because Polygon K9L9M9N9 is the image of Polygon KLMN after a translation, each of its sides is congruent to the corresponding side of Polygon KLMN, and each of its angles is congruent to the corresponding angle of Polygon KLMN

/ K > / K9 / L > / L9 / M > / M9 / N > / N9

··· KL ù ··· K9L9 ··· LM ù ··· L9M9 ··· MN ù ··· M9N9 ··· NK ù ··· N9K9

All of the corresponding parts are congruent, so the polygons are congruent

x

y

K N

ML

M9

O

L9

K9 N9

Vocabularycongruent polygons polygons with exactly

the same size and shape

The symbol ù is read “is

congruent to ”

B C

A

D

E F

nABC ù nDEF

211211

They did not change. Possible explanation: The

lengths of the sides and the measures of the

angles stay the same in a translation.

B

B

The measures would be the same.

©Curriculum Associates, LLC Copying is not permitted.212 Lesson 20 Transformations and Similarity

3 Triangle ABC and its image are shown

a. What type of transformation was used to transform nABC to nA9B9C9?

b. Is nA9B9C9 congruent to nABC? Explain why or why not

4 Consider Triangle D and Triangle X

a. Is Triangle X the result of a refl ection, translation, or rotation of Triangle D? Explain how you know

b. Are the triangles congruent? Explain why or why not

5 Polygon A was translated 7 units to the left to form Polygon R Name another way to transform Polygon A to form Polygon R

6 Polygon P is refl ected to form Polygon S Sasha says that the perimeter of Polygon S is the same as the perimeter of Polygon P Do you agree with Sasha? Explain why or why not

Solve.

x

y

B

CA

A9 C9

B9

O23 3

2

x

y

D X

O2224 2 4 6

2

x

y

AR

O2224 2 4 6

2

212

M

M

M

C

a reflection over the line y 5 2

Yes; Possible explanation: The lengths of the sides and the measures of the angles stay

the same in a reflection, so the triangles are congruent.

No, Triangle X is not a reflection, translation, or

rotation of Triangle D because it is not the

same size or shape as Triangle D.

No, the triangles are not congruent because not all of the angles and sides in

Triangle D are congruent to the angles and sides in Triangle X.

Possible answer: Reflect Polygon A over the line x 5 1.

I agree with Sasha. Possible explanation: When a figure is reflected, the lengths of the

corresponding sides of the polygons are congruent. So the perimeters are the same.

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Unit 4 G

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Unit 4

Practice Lesson 20 Transformations and Sim

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©Curriculum Associates, LLC Copying is not permitted. 213Lesson 20 Transformations and Similarity

Name:

Combine Dilations and Other Transformations

Study the example problem showing how to combine a dilation with other transformations. Then solve problems 1–6.

1 Suppose the scale factor of the dilation in the example

was 2 instead of 1 ·· 2 , but the dilation was still centered

about O and nABC was still rotated 180° about O What

would the coordinates of the vertices of nHJK be?

H( ) J( ) K( )

2 Explain how a dilation is diff erent from a translation, a refl ection, or a rotation

Example

In the diagram, nABC is similar to nHJK A sequence of transformations was used to transform nABC to nHJK

Describe the change in the coordinates

A(2, 4) was transformed to H(21, 22)

B(6, 22) was transformed to J(23, 1)

C(2, 22) was transformed to K(21, 1)

Each x-coordinate has the opposite sign and was multiplied by 1 ·· 2

Each y-coordinate has the opposite sign and was multiplied by 1 ·· 2

nABC was dilated about center O with a scale factor of 1 ·· 2 and rotated

180° about O.

Lesson 20

Ox

y

A

BCH

J K

Vocabularydilation a

transformation in which

the original figure and

the image are similar

scale factor in a

dilation, the ratio of the

lengths of corresponding

sides of the figure and its

image

center the center of a

dilation is the point that

is transformed onto itself

by the dilation

213

A dilation can change the size of a figure.

Translations, reflections, and rotations do not

change the size.

24, 28 212, 4 24, 4

B

B

©Curriculum Associates, LLC Copying is not permitted.214 Lesson 20 Transformations and Similarity

Solve.

3 The coordinates of the vertices of Polygon RSTV are

R(2, 4), S(6, 4), T(6, 0), and V(2, 0) The Polygon is dilated with

scale factor of 3 ·· 2 and center (0, 0) Explain how you can fi nd

the coordinates of the vertices of Polygon R9S9T9V9 from

the coordinates of the vertices of the Polygon RSTV

4 Triangle PQR is shown at the right

a. Reflect nPQR across the y-axis and then dilate it about center O with a scale factor of 2 Sketch the final image

b. Compare the coordinates of the corresponding vertices of the final image and nPQR.

5 In the diagram at the right, Polygon A is similar to Polygon W What sequence of transformations transformed Polygon A to Polygon W?

6 Tracy dilates a fi gure with a scale factor of 3 ·· 4 and center O

and then dilates the image with a scale factor of 2 and center O Carrie says that she can get the same fi nal image using just one dilation Is she correct? If so, how can she do that? If not, why not?

x

y

P Q

R O

x

y

A

W

O

214

Possible explanation: I can multiply the x-coordinate and the y-coordinate of each vertex

by 3 ·· 2 . R9(3, 6), S9(9, 6), T9(9, 0), V9(3, 0)

M

M

M

Possible answer: Each of the image coordinates is twice the corresponding coordinate

in nPQR. Each x-coordinate in the image has the opposite sign of the corresponding

x-coordinate in nPQR.

C

Possible answer: Translation 2 units down and

6 units to the right followed by dilation with

center O and scale factor 2 ·· 3 .

Yes; Possible answer: She can dilate the figure with a scale factor of 3 ·· 4 • 2 5 3 ·· 2 and

center O.

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Unit 4 G

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nit 4Practice Lesson 20 Transform

ations and Similarity

©Curriculum Associates, LLC Copying is not permitted. 215Lesson 20 Transformations and Similarity

Name:

1 Polygon ABCD is shown on the coordinate plane Sketch the image after it is rotated 90° clockwise about O and then dilated with scale factor 2 and center O.

Transformations and Similarity

Solve the problems.

Make sure you rotate the polygon clockwise.

Lesson 20

x

y

A B

CD

O

2 The coordinates of nDEF are D(24, 4), E(2, 4), and F(0, 2)

The triangle is dilated with scale factor 1 ·· 2 and center O

What are the coordinates of the vertices of the image

of nDEF?

A (2, 22), (21, 22), (0, 21)

B (28, 8), (4, 8), (0, 4)

C (22, 2), (1, 2), (0, 1)

D (4, 24), (4, 2), (2, 0)

Sue chose A as the correct answer How did she get that answer?

How do you use the scale factor to find the coordinates of the image?

215

M

B

Sue thought that the dilation also changed the signs of the x- and

y-coordinates.

©Curriculum Associates, LLC Copying is not permitted.216 Lesson 20 Transformations and Similarity

Solve.

3 Tell whether each statement is True or False

a. A dilation image is always congruent to the original figure u True u False

b. A rotation image is always congruent to the original figure u True u False

c. A reflection image is never congruent to the original figure u True u False

d. A translation image is always congruent to the original figure u True u False

4 Polygon LMNP was transformed to Polygon WXYZ

Part ADescribe a sequence of transformations that maps Polygon LMNP to Polygon WXYZ

Part BFind the perimeters of Polygon WXYZ and Polygon LMNP Then write the ratio of the perimeter of Polygon WXYZ to the perimeter of Polygon LMNP. How does this ratio compare to the scale factor you found in Part A?

What types of transformations keep the size and shape of the original figure?

What type of transformation can change the size of a figure?

x

y

Z

WL M

NP

Y

X

O22 2

22

3

216

Possible answer: A dilation with a scale factor of 2 ·· 3

and center O and a reflection in the line y 5 1

M

C

Perimeter of polygon WXYZ: 16 units;

perimeter of polygon LMNP: 24 units;

ratio: 16 ··· 24 5 2 ·· 3 ; The ratio of the perimeters is

the same as the scale factor.

3

3

3

3