Similarity TransformationsLESSON 7–6

Targeted TEKSG.7(A) Apply the definition of similarity in terms of a dilation toidentify similar figures and their proportional sides and the congruent corresponding angles.

Mathematical ProcessesG.1(E), G.1(G)

• Identify similarity transformations.

• Verify similarity after a similarity transformation.

I will

• dilation

• similarity transformation

• center of dilation

• scale factor of a dilation

• enlargement

• reduction

Scale Factor -- k

Example 1A

Identify a Dilation and Find Its Scale Factor

A. Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation.

B is smaller than A, so the dilation is a reduction.

The distance between the vertices at (2, 2) and (2, –2) for

A is 4 and from the vertices at (1, 1) and (1, –1) for B is 2.

Answer: So, the scale factor is or . __12

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Example 1B

Identify a Dilation and Find Its Scale Factor

B. Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation.

B is larger than A, so the dilation is an enlargement.

The distance between the vertices at (3, 3) and (–3, 3) for A is 6 and from the vertices at (1, 1) and (–1, 1) for B is 2.

Answer: So, the scale factor is or 3. __62

Example of Determining Scale Factor

A. Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation.

A. reduction;

B. reduction;

C. enlargement; 2

D. enlargement; 3

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Example 1 CYP B

B. Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation.A. reduction;

B. reduction;

C. enlargement;

D. enlargement; 2

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Find and Use a Scale Factor

PHOTOCOPYING A photocopy of a receipt is 1.5 inches wide and 4 inches long. By what percent should the receipt be enlarged so that its image is 2 times the original? What will be the dimensions of the enlarged image?

To enlarge the receipt 2 times the original, use a scale factor of 2. Written as a percent, the scale factor is (2 ● 100%) or 200%. Now, find the dimensions of the enlarged receipt.width: 1.5 in. ● 200% = 3 in. length: 4 in. ● 200% = 8 in.

The enlarged receipt will be 3 inches by 8 inches.

A. 15 inches by 25 inches

B. 8 inches by 15 inches

C. 12 inches by 22.5 inches

D. 16 inches by 30 inches

PHOTOGRAPHS Mariano wants to enlarge a picture he took that is 4 inches by 7.5 inches. He wants it to fit perfectly into a frame that is 400% of the original size. What will be the dimensions of the enlarged photo?

Example 3A

Verify Similarity after a Dilation

A. Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation.

original: A(–6, –3), B(6, –3), C(–6, 6)

image: D(–2, –1), E(2, –1), F(–2, 2)

Graph each figure. Since ∠A and ∠D are both right angles, ∠A ≅ ∠D Show that the lengths of the sides that include ∠A and ∠D are proportional.

Example 3A

Verify Similarity after a Dilation

Answer: Since the lengths of the sides that include ∠M and ∠D are proportional, ΔMNO ~ ΔDFG by SAS Similarity.

Verify Similarity after a Dilation

B. Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation.

original: P(2, 1), Q(4, 1), S(2, 0), R(4, 0)image: W(4, 2), X(8, 2), R(4, 0), T(8, 0)

Verify Similarity after a Dilation

Since the figures are rectangles, their corresponding angles are congruent.

1. Dilation are performed with respect to a ____________ called the center of dilation.

2. A dilation is a transformation that _________ or ________ that original figures proportionally.

3. A dilation with a scale factor __________ produces an enlargement.

4. A scale factor of a dilation describes the ________of the dilation.

5. A similarity transformation is a transformation that maps a figure onto a ________ figure.

6. A dilation with a scale factor ___________ produces a reduction.

1. Dilation are performed with respect to a ____________ called the center of dilation.

2. A dilation is a transformation that _________ or ________ that original figures proportionally.

3. A dilation with a scale factor __________ produces an enlargement.

4. A scale factor of a dilation describes the ________of the dilation.

5. A similarity transformation is a transformation that maps a figure onto a ________ figure.

6. A dilation with a scale factor ___________ produces a reduction.

fixed point

enlarges reduces

extent

similar

A. Graph the original figure and its dilated image. Then determine the scale factor of the dilation.original: B(–7, –2), A(5, –2), D(–7, 7)image: J(–3, 0), K(1, 0), L(–3, 3)

A.

B.

C.

D.

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B. Graph the original figure and its dilated image. Then determine the scale factor of the dilation.original: A(4, 3), B(6, 3), C(4, 2), D(6, 2)image: E(6, 4), F(10, 4), G(6, 2), H(10, 2)A. 2

B.

C. 3

D. 4

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Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation.

A. enlargement;

B. enlargement;

C. reduction;

D. reduction;

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Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation.

A. enlargement;

B. enlargement;

C. reduction;

D. reduction;

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The coordinates of the vertices of two triangles are listed in the table. What is the scale factor of the dilation from ΔABC to ΔXYZ?A.

B.

C. 2

D. 3

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Similarity TransformationLESSON 7–6