Holt Geometry

12-7 Dilations12-7 Dilations

Holt Geometry

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Geometry

12-7 Dilations

Warm Up

1. Translate the triangle with vertices A(2, –1), B(4, 3), and C(–5, 4) along the vector <2, 2>.

2. ∆ABC ~ ∆JKL. Find the value of JK.

A'(4,1), B'(6, 5),C(–3, 6)

Holt Geometry

12-7 Dilations

Identify and draw dilations.

Objective

Holt Geometry

12-7 Dilations

center of dilationenlargementreduction

Vocabulary

Holt Geometry

12-7 Dilations

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

Holt Geometry

12-7 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 Geometry

12-7 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 Geometry

12-7 Dilations

For a dilation with scale factor k, if k > 0, the figure is not turned or flipped. If k < 0, the figure is rotated by 180°.

Helpful Hint

Holt Geometry

12-7 Dilations

Holt Geometry

12-7 Dilations

A dilation enlarges or reduces all dimensions proportionally. A dilation with a scale factor greater than 1 is an enlargement, or expansion. A dilation with a scale factor greater than 0 but less than 1 is a reduction, or contraction.

Holt Geometry

12-7 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 Geometry

12-7 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 Geometry

12-7 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 Geometry

12-7 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 Geometry

12-7 Dilations

Holt Geometry

12-7 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 Geometry

12-7 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 Geometry

12-7 Dilations

Example 4 Continued

Graph the preimage and image.

P’

Q’R’

P

R

Q

Holt Geometry

12-7 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 Geometry

12-7 Dilations

Check It Out! Example 4 Continued

Graph the preimage and image.

RS

TU

R’

S’T’ U’

Holt Geometry

12-7 Dilations

Lesson Quiz: Part I

1. Tell whether the transformation appears to be a dilation.

yes

2. Copy ∆RST and the center of dilation. Draw the image of ∆RST under a dilation with a scale of .

Holt Geometry

12-7 Dilations

3. A rectangle on a transparency has length 6cm and width 4 cm and with 4 cm. On the transparency 1 cm represents 12 cm on the projection. Find the perimeter of the rectangle in the projection.

Lesson Quiz: Part II

4. Draw the image of the triangle with vertices E(2, 1), F(1, 2), and G(–2, 2) under a dilation with a scale factor of –2 centered at the origin.

240 cm