Holt McDougal Geometry

9-7 Dilations

A dilation is a transformation that changes the size of a figure but not its shape. The preimage and the image are always similar.

AA A’A’

Holt McDougal Geometry

9-7 Dilations

Example 1: Identifying Dilations

Tell whether each transformation appears to be a dilation. Explain.

A. B.

Yes; the figures are similar and the image is not turned or flipped.

No; the figures are not similar.

Holt McDougal Geometry

9-7 Dilations

Check It Out! Example 1

a. b.

Yes, the figures are similar and the image is not turned or flipped.

No, the figures are not similar.

Tell whether each transformation appears to be a dilation. Explain.

Holt McDougal Geometry

9-7 Dilations

A dilation, or similarity transformation, is a transformation in which every point P and its image P’ have the same ratio.

Center ofdilation

CP PP’ PQCQ QQ’ P’Q’

K =

Holt McDougal Geometry

9-7 Dilations

k > 1 is an enlargement, or expansion.

0< k < 1 is a reduction, or contraction.

A scale factor describes how much the figure is enlarged or reduced.

For a dilation with scale factor k, you can find the image of a point by multiplying each coordinate by k: (a, b) (ka, kb).

Holt McDougal Geometry

9-7 Dilations

Example 2: Drawing Dilations

Copy the figure and the center of dilation P. Draw the image of ∆WXYZ under a dilation with a scale factor of 2.

Step 1 Draw a line through P and each vertex.

Step 2 On each line, mark twice the distance from P to the vertex.

Step 3 Connect the vertices of the image.

W’ X’

Z’Y’

Holt McDougal Geometry

9-7 Dilations

Check It Out! Example 2

Copy the figure and the center of dilation. Draw the dilation of RSTU using center Q and a scale factor of 3.

Step 1 Draw a line through Q and each vertex.

Step 2 On each line, mark twice the distance from Q to the vertex.

Step 3 Connect the vertices of the image.

R’ S’

T’U’

Holt McDougal Geometry

9-7 Dilations

Example 1: Drawing and Describing Dilations

D: (x, y) → (3x, 3y) A(1, 1), B(3, 1), C(3, 2)

A’ (3, 3), B’ (9, 3), C’ (9,6) scale factor 3

A. Apply the dilation D to the polygon with the given vertices. Describe the dilation.

Holt McDougal Geometry

9-7 Dilations

Example 1: Continued

P’(-6, 3), Q’ (-3, 6), R’ (3, 3) scale factor 3/4

D: (x, y) →

P(–8, 4), Q(–4, 8), R(4, 4)

43 x,

43 y

B. Apply the dilation D to the polygon with the given vertices. Describe the dilation.

Holt McDougal Geometry

9-7 Dilations

Check It Out! Example 1

D'(-2, 0), E'(-2, -1), F'(-1, -2); scale factor 1/4

Name the coordinates of the image points. Describe the dilation.

(x, y)→ ( ¼ x, ¼ y)

D(-8, 0), E(-8, -4), and F(-4, -8).

Holt McDougal Geometry

9-7 Dilations

Example 3: Drawing Dilations

On a sketch of a flower, 4 in. represent 1 in. on the actual flower. If the flower has a 3 in. diameter in the sketch, find the diameter of the actual flower.

The scale factor in the dilation is 4, so a 1 in. by 1 in. square of the actual flower is represented by a 4 in. by 4 in. square on the sketch.

Let the actual diameter of the flower be d in.

3 = 4d

d = 0.75 in.

Holt McDougal Geometry

9-7 Dilations

Check It Out! Example 3

What if…? An artist is creating a large painting from a photograph into square and dilating each square by a factor of 4. Suppose the photograph is a square with sides of length 10 in. Find the area of the painting.

The scale factor of the dilation is 4, so a 10 in. by 10 in. square on the photograph represents a 40 in. by 40 in. square on the painting.

Find the area of the painting.

A = l w = 4(10) 4(10)

= 40 40 = 1600 in2

Holt McDougal Geometry

9-7 Dilations

If the scale factor of a dilation is negative, the preimage is rotated by 180°.

For k > 0, a dilation with a scale factor of –k is equivalent to the composition of a dilation with a scale factor of k that is rotated 180° about the center of dilation.

Holt McDougal Geometry

9-7 Dilations

Example 4: Drawing Dilations in the Coordinate Plane

Draw the image of the triangle with vertices P(–4, 4), Q(–2, –2), and R(4, 0) under a

dilation with a scale factor of centered at the origin.

The dilation of (x, y) is

Holt McDougal Geometry

9-7 Dilations

Example 4 Continued

Graph the preimage and image.

P’

Q’R’

P

R

Q

Holt McDougal Geometry

9-7 Dilations

Check It Out! Example 4 Draw the image of the triangle with vertices R(0, 0), S(4, 0), T(2, -2), and U(–2, –2) under a dilation centered at the origin with a scale factor of .

The dilation of (x, y) is

Holt McDougal Geometry

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Check It Out! Example 4 Continued

Graph the preimage and image.

RS

TU

R’S’T’ U’

Holt McDougal Geometry

9-7 Dilations

A(–6, -6), B(-6, 3), C(3, 3), D(3, -6)

H(-2, -2), J(-2, 1), K(1, 1), L(1, -2)

ABCD maps to HJKL

(x, y) → 13

x 13

y,

Determine whether the polygons are similar.

Holt McDougal Geometry

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P(2, 0), Q(2, 4), R(4, 4),S(4, 0)

W(5, 0), X(5, 10), Y(8, 10), Z(8, 0).

Determine whether the polygons are similar.

No; (x, y) → (2.5x, 2.5y) maps P to W, but not S to Z.

Holt McDougal Geometry

9-7 Dilations

A(1, 2), B(2, 2), C(1, 4) D(4, -6), E(6, -6), F(4, -2)

Determine whether the polygons are similar.

Yes;

translation: (x, y) → (x + 1, y - 5).

dilation: (x, y) → (2x, 2y).

Holt McDougal Geometry

9-7 Dilations

F(3, 3), G(3, 6), H(9, 3), J(9, –3)

S(–1, 1), T(–1, 2), U(–3, 1), V(–3, –1)

Determine whether the polygons are similar.

Yes; reflection: (x, y) → (-x, y).

dilation: (x, y) → (1/3 x, 1/3 y)

Holt McDougal Geometry

9-7 Dilations

A(2, -1), B(3, -1), C(3, -4)

P(3, 6), Q(3, 9), R(12, 9).

Determine whether the polygons are similar.

Yes;

rotation: (x, y) → (-y, x)

dilation: (x, y) → (3x, 3y)

Holt McDougal Geometry

9-7 Dilations

Determine whether the polygons are similar. If so, describe the transformation in 2 different ways, from the larger to the smaller, and the smaller to the larger.

Holt McDougal Geometry

9-7 Dilations

Determine whether the polygons are similar. If so, describe the transformation in 2 different ways, from the larger to the smaller, and the smaller to the larger.

Holt McDougal Geometry

9-7 Dilations

Example 3: Proving Triangles Are Similar

Prove: ∆EHJ ~ ∆EFG.

Given: E(–2, –6), F(–3, –2), G(2, –2), H(–4, 2), and J(6, 2).

Step 1 Plot the points and draw the triangles.

Holt McDougal Geometry

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Example 3 Continued

Step 2 Use the Distance Formula to find the side lengths.

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Example 3 Continued

Step 3 Find the similarity ratio.

= 2 = 2

Since and E E, by the Reflexive Property, ∆EHJ

~ ∆EFG by SAS ~ .

Holt McDougal Geometry

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Check It Out! Example 3

Given: R(–2, 0), S(–3, 1), T(0, 1), U(–5, 3), and V(4, 3).

Prove: ∆RST ~ ∆RUV

Holt McDougal Geometry

9-7 Dilations

Check It Out! Example 3 Continued

Step 1 Plot the points and draw the triangles.

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-1

1

2

3

4

5

X

Y

R

ST

UV

Holt McDougal Geometry

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Check It Out! Example 3 Continued

Step 2 Use the Distance Formula to find the side lengths.

Holt McDougal Geometry

9-7 Dilations

Check It Out! Example 3 Continued

Step 3 Find the similarity ratio.

Since and R R, by the Reflexive Property,

∆RST ~ ∆RUV by SAS ~ .