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Problem 1

594 Chapter 9 Transformations

Similarity Transformations

9-7

Objectives To identify similarity transformations and verify properties of similarity

Your friend says that she performed a composition of transformations to map △ABC to △A′B′C′. Describe the composition of transformations. A

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Lesson Vocabulary

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LessonVocabulary

Is there more than one composition of transformations possible to map △ABC to △A′B′C′?

In the Solve It, you used a composition of a rigid motion and a dilation to describe the mapping from △ABC to △A′B′C′.

Essential Understanding You can use compositions of rigid motions and dilations to help you understand the properties of similarity.

Drawing Transformations

△DEF has vertices D(2, 0), E(1, 4), and F(4, 2). What is the image of △DEF when you apply the composition D1.5 ∘ Ry@axis?

Step 1 Find the vertices of Ry@axis(△DEF). Then connect the vertices to draw the image.

Ry@axis (D) = D′(-2, 0) Ry@axis (E) = E′(-1, 4) Ry@axis (F) = F′(-4, 2)

Step 2 Find the vertices of the dilation of △D′E′F′. Then connect the vertices to draw the image.

D1.5 (D′) = D″(-3, 0) D1.5 (E′) = E″(-1.5, 6) D1.5 (F′) = F″(-6, 3)

The vertices of the image after the composition of transformations are D″(-3, 0), E″(-1.5, 6), and F″(-6, 3).

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G-SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar . . . Also G-SRT.A.3

MP 1, MP 2, MP 3, MP 4

MATHEMATICAL PRACTICES

Common Core State Standards

Problem 2

Lesson 9-7 SimilarityTransformations 595

1. Reasoning △LMN has vertices L(-4, 2), M(-3, -3), and N(-1, 1). Suppose the triangle is translated 4 units right and 2 units up and then dilated by a scale factor of 0.5 with center of dilation at the origin. Sketch the resulting image of the composition of transformations.

Describing Transformations

What is a composition of rigid motions and a dilation that maps △RST to △PYZ?

Study the figures to determine how the image could have resulted from the preimage. Then use the vertices to verify the composition of transformations.

A composition of transformations that maps △RST to △PYZ

The vertices of the preimage and image

It appears that △RST was rotated and then enlarged to create △PYZ. To verify the composition of transformations, begin by rotating the triangle 180° about the origin.

r(180°, O) (R) = R′(-1, -1) Use the rule r(180°, O)(x, y) = (-x, -y).

r(180°, O) (S) = S′(-1, -3)

r(180°, O) (T) = T ′(-3, -1)

△PYZ appears to be about twice as large as △RST. Scale the vertices of the intermediate image R′S′T ′ to verify the composition.

D2(-1, -1) = P(-2, -2) Use the rule D2(x, y) = (2x, 2y).

D2(-1, -3) = Y(-2, -6)

D2(-3, -1) = Z(-6, -2)

The vertices of the dilation of △R′S′T′ match the vertices of △PYZ.

A rotation of 180° about the origin followed by a dilation with scale factor 2 maps △RST to △PYZ.

2. What is a composition of rigid motions and a dilation that maps trapezoid ABCD to trapezoid MNHP?

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Problem 3


Finding Similarity Transformations

Is there a similarity transformation that maps △PAQ to △TNO? If so, identify the similarity transformation and write a similarity statement. If not, explain.

Although PA ≠ TN, there is a scale factor k such that k # PA = TN. Dilate △PAQ using this scale factor. Then

P′A′ ≅ TN. Since dilations preserve angle measure, you also know that ∠P′ ≅ ∠T and ∠A′ ≅ ∠N. Therefore, △P′A′Q′ ≅ △TNO by ASA. This means that there is a sequence of rigid motions that maps △P′A′Q′ onto △TNO.

So, there is a dilation that maps △PAQ to △P′A′Q′, and a sequence of rigid motions that maps △P′A′Q′ to △TNO. Therefore, there is a composition of a dilation and rigid motions that maps △PAQ onto △TNO.

3. Is there a similarity transformation that maps △JKL to △RST ? If so, identify the similarity transformation and write a similarity statement. If not, explain.

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Here’s Why It Works Consider the composition of a rigid motion and a dilation shown at the right.

Because rigid motions and dilations preserve angle measure, m∠P = m∠P′, m∠Q = m∠Q′, and m∠R = m∠R′. So, corresponding angles are congruent.

Because there is a dilation, there is some scale factor k such that:

PQ = kP′Q′ QR = kQ′R′ PR = kP′R′

k = PQ

P′Q′ k = QR

Q′R′ k = PRP′R′

So, PQ

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Does it matter what the center of dilation is? No. All that matters is that k # PA = TN.

Notice that the figures in Problems 1 and 2 appear to have the same shape but different sizes. Compositions of rigid motions and dilations map preimages to similar images. For this reason, they are called similarity transformations. Similarity transformations give you another way to think about similarity.

Key Concept Similar Figures

Two figures are similar if and only if there is a similarity transformation that maps one figure onto the other.

Lesson Check

Problem 4


How can you determine whether two figures are similar if you have no information about side lengths or angle measures?Any two plane figures are similar if you can find a similarity transformation that maps one onto the other.

Similarity transformations provide a powerful general approach to similarity. In Problem 3, you used similarity transformations to verify the AA Postulate for triangle similarity. Another advantage to the transformational approach to similarity is that you can apply it to figures other than polygons.

Determining Similarity

A new company is using a computer program to design its logo. Are the two figures used in the logo so far similar?

If you can find a similarity transformation between two figures, then you know they are similar. The smaller lightning bolt can be translated so that the tips coincide. Then it can be enlarged by some scale factor so that the two bolts overlap.

The figures are similar because there is a similarity transformation that maps one figure onto the other. The transformation is a translation followed by a dilation.

4. Are the figures at the right similar? Explain.Got It?

Do you know HOW?Use the diagram below for Exercises 1 and 2.

1. What is a similarity transformation that maps △RST to △JKL?

2. What are the coordinates of (D14

∘ r(180°, O))(△RST)?

Do you UNDERSTAND? 3. Vobabulary Describe how the word dilation is

used in areas outside of mathematics. How do these applications relate the mathematical definition?

4. Open Ended For △TUV at the right, give the vertices of a similar triangle after a similarity transformation that uses at least 1 rigid motion.

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Practice and Problem-Solving Exercises

△MAT has vertices M(6, −2), A(4, −5), and T(1, −2). For each of the following, sketch the image of the composition of transformations.

5. reflection across the x-axis followed by a dilation by a scale factor of 0.5

6. rotation of 180° about the origin followed by a dilation by a scale factor of 1.5

7. translation 6 units up followed by a reflection across the y-axis and then a dilation by a scale factor of 2

For each graph, describe the composition of transformations that maps △FGH to △QRS.

8. 9. 10.

For each pair of figures, determine if there is a similarity transformation that maps one figure onto the other. If so, identify the similarity transformation and write a similarity statement. If not, explain.

11. 12. 13.

Determine whether or not each pair of figures below is similar. Explain your reasoning.

14. 15.

PracticeA See Problem 1.

See Problem 2.

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16. Writing Your teacher uses geometry software program to plot △ABC with vertices A(2, 1), B(6, 1), and C(6, 4). Then he used a similarity transformation to plot △DEF with vertices D(-4, -2), E(-12, -2), and F(-12, -8). The corresponding angles of the two triangles are congruent. How can the Distance Formula be used to verify that the ratios of the corresponding sides are proportional? Verify that the figures are similar.

17. Think About a Plan Suppose that △JKL is formed by connecting the midpoints of △ABC. Is △AJL similar to △ABC? Explain.

• How are the side lengths of △AJL related to the side lengths of △ABC ?

• Can you find a similarity transformation that maps △AJL to △ABC ? Explain.

18. Writing What properties are preserved by rigid motions but not by similarity transformations?

Determine whether each statement is always, sometimes, or never true.

19. There is a similarity transformation between two rectangles.

20. There is a similarity transformation between two squares.

21. There is a similarity transformation between two circles.

22. There is a similarity transformation between a right triangle and an equilateral triangle.

23. Indirect Measurement A surveyor wants to use similar triangles to determine the distance across a lake as shown at the right.

a. Are the two triangles in the figure similar? Justify your reasoning.

b. What is the distance d across the lake?

24. Photography A 4-inch by 6-inch rectangular photo is enlarged to fit an 8-inch by 10-inch frame. Are the two photographs similar? Explain.

25. Reasoning Is a rigid motion an example of a similarity transformation? Explain your reasoning and give an example.

26. Art A printing company enlarges a banner for a graduation party by a scale factor of 8.

a. What are the dimensions of the larger banner? b. How can the printing company be sure that the

enlarged banner is similar to the original?

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27. If △ABC has vertices given by A(u, v), B(w, x), and C(y, z), and △NOP has vertices given by N(5u, -4v), O(5w, -4x), and P(5y, -4z), is there a similarity transformation that maps △ABC to △NOP? Explain.

28. Overhead Projector When Mrs. Sheldon places a transparency on the screen of the overhead projector, the projector shows an enlargement of the transparency on the wall. Does this situation represent a similarity transformation? Explain.

29. Reasoning Tell whether each statement below is true or false. a. In order to show that two figures are similar, it is sufficient to show that there is a

similarity transformation that maps one figure to the other. b. If there is a similarity transformation that maps one figure to another figure, then

the figures are similar. c. If there is a similarity transformation that maps one figure to another figure, then

the figures are congruent.

ChallengeC

Mixed Review 34. Which capital letters of the alphabet are rotation images of themselves?

Draw each letter and give an angle of rotation (6 360°) .

35. Three vertices of an isosceles trapezoid are (-2, 1), (1, 4), and (4, 4). Find all possible coordinates for the fourth vertex.

Get Ready! To prepare for Lesson 10-1, do Exercises 34–37.

Find the area of each figure.

36. a square with 5-cm sides 37. a rectangle with base 4 in. and height 7 in.

38. a 4.6 m-by-2.5 m rectangle 39. a rectangle with length 3 ft and width 12 ft

See Lesson 9-3.

See Lesson 6-7.

See Lesson 1-8.

Standardized Test Prep

30. △STU has vertices S(1, 2), T(0, 5), and U(-8, 0). What is the x-coordinate of S after a 270° rotation about the origin?

31. The diagonals of rectangle PQRS intersect at O. PO = 2x - 5 and OR = 7 - x.

What is the length of QS?

32. The length of the hypotenuse of a 45°-45°-90° triangle is 55 in. What is the length of one of its legs to the nearest tenth of an inch?

33. You place a sprinkler so that it is equidistant from three rose bushes at points A, B, and C. How many feet is the sprinkler from A?

SAT/ACT

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