Dilations

Warm Up

Lesson PresentationProblem of the Day

Lesson Quizzes

Dilations

Warm UpMultiply.

1. 4 • 2. 12 •

3. 24 • 4. –36 •

3

18

10

9

–27

30

3 4

3 4

3 4

3 4

5. 4 • 2.5 6. 12 • 2.5

Dilations

Problem of the Day

Every day, a plant grows to three times its size. Every night, it shrinks to half its size. After three days and nights, it is 6.75 in. tall. How tall was the plant at the start? 2 in.

Dilations

Learn to identify and create dilations of plane figures.

Dilations

Vocabularydilationcenter of dilationscale factor

Dilations

Your pupils are the black areas in the center of your eyes. When you go to the eye doctor, the doctor may dilate your pupils, which makes them larger.

A dilation is a transformation that changes the size, but not the shape, of a figure.

Dilations

Every dilation has a fixed point that is the center of dilation. To find the center of dilation, draw a line that connects each pair of corresponding vertices. The lines intersect at one point. This point is the center of dilation.

Dilations

Tell whether each transformation is a dilation. Explain. Compare the ratios of

corresponding side lengths.

Additional Example 1A: Identifying Dilations

Yes; the ratios of corresponding side lengthare equal and corresponding angles are congruent.

26

13

BCB'C' = =

26

13

ADA'D' = =

Dilations

Tell whether each transformation is a dilation. Explain. Compare the ratios of

corresponding side lengths.

Additional Example 1B: Identifying Dilations

No; the ratios of corresponding side length are not equal.

42

BAB'A' = = 2

53

ACA'C' = 1.66=

Dilations

Tell whether each transformation is a dilation. Explain. Compare the ratios of

corresponding side lengths.

Check It Out: Example 1

No, the ratios are not equal, so the triangles are not similar.

36

XYX‘Y' = = 0.5

34

YZY‘Z' = = 0.75

Dilations

The scale factor describes how much a figure is enlarged or reduced. It represents the ratio of a length on the image to the corresponding lentth on the original figure.

DilationsAdditional Example 2A: Using the Origin as the

Center of DilationDilate the figure below by a scale factor of 2. What are the vertices of the image?

Multiply the coordinates by 2 to find the vertices of the image.

A(2, 2) A’(2 • 2, 2 • 2) A’(4, 4)

B(3, 4) B’(3 • 2, 4 • 2) B’(6, 8)

C(5, 2) C’(5 • 2, 2 • 2) C’(10, 4)

The vertices of the image are A’(4, 4), B’(6, 8), and C’(10, 4).

ABC A’B’C’

DilationsAdditional Example 2B: Using the Origin as the

Center of Dilation

Dilate the figure below by a scale factor of . What are the vertices of the image?

1 3

The vertices of the image are A’(1, 3), B’(3, 2), and C’(2, 1).

ABC A’B’C’

A(3, 9) A’(3 • , 9 • ) A’(1, 3)1 3

1 3

B(9, 6) B’(9 • , 6 • ) B’(3, 2)1 3

1 3

C(6, 3) C’(6 • , 3 • ) C’(2, 1)1 3

1 3

Multiply the coordinates by to find the vertices of the image.

13

Dilations

Check It Out: Example 2A

Dilate the figure by a scale factor of 2. What are the vertices of the image?

2 4 6 8 100

2

4

6

8

10

B

C

A

Dilations

Check It Out: Example 2A Continued

A(2, 2) A’(2 • 2, 2 • 2) A’(4, 4)

B(4, 2) B’(4 • 2, 2 • 2) B’(8, 4)

C(2, 4) C’(2 • 2, 4 • 2) C’(4, 8)

ABC A’B’C’

The vertices of the image are A’(4, 4), B’(8, 4), and C’(4, 8).

Dilations

Check It Out: Example 2A Continued

2

4

2 4 6 8 100

6

8

10

B’

C’

A’

B

C

A

Dilations

Check It Out: Example 2B

Dilate the figure by a scale factor of 0.5. What are the vertices of the image?

2

4

2 4 6 8 100

6

8

10

B

C

A

Dilations

Check It Out: Example 2B Continued

A(4, 5) A’(4 • 0.5, 5 • 0.5) A’(2, 2.5)

B(8, 5) B’(8 • 0.5, 5 • 0.5) B’(4, 2.5)

C(4, 9) C’(4 • 0.5, 9 • 0.5) C’(2, 4.5)

ABC A’B’C’

The vertices of the image are A’(2, 2.5), B’(4, 2.5), and C’(2, 4.5).

Dilations

Check It Out: Example 2B Continued

2

4

2 4 6 8 100

6

8

10

B

C

A

B’

C’

A’

Dilations

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

Dilations

Lesson Quiz1. Tell whether the transformation is a dilation. Explain.

2. A triangle has vertices A'(2,4), B(5,6), and C(6,1). Dilate the figure by a scale factor of 2 with the origin as the center of dilation. What are the vertices of the image? A’(4,8) B’(10,12) C’(12,2)

Yes, the ratios of corresponding side lengths are equal.

Dilations

1. Identify the transformation that is a dilation.

A. A(0, 3), B(4, 4), C(2, 2) à A’(0, 9), B’(12, 12), C’(6, 6)

B. A(2, 3), B(3, 5), C(5, 7) à A’(5, 6), B’(6, 8), C’(8, 10)

C. A(5, 2), B(3, 7), C(7, 9) à A’(2, –1), B’(0, 4), C’(4, 6)

D. A(1, 2), B(3, 4), C(5, 6) à A’(4, 5), B’(6, 7), C’(8, 9)

Lesson Quiz for Student Response Systems