Holt Geometry

12-7 Dilations12-7 Dilations

Holt Geometry

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Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Geometry

12-7 Dilations

Are you ready?

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.

Holt Geometry

12-7 Dilations

TSW identify and draw dilations.

Objective

Holt Geometry

12-7 Dilations

center of dilationenlargementreduction

Vocabulary

Holt Geometry

12-7 Dilations

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.

Holt Geometry

12-7 Dilations

Example 2

a. b.

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 3: 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.

Holt Geometry

12-7 Dilations

Example 3: Drawing Dilations

You should get something that looks like this:

W’ X’

Z’Y’

Holt Geometry

12-7 Dilations

Example 4

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.

Holt Geometry

12-7 Dilations

Example 4

R’ S’

T’U’

Holt Geometry

12-7 Dilations

Example 5: 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.

Holt Geometry

12-7 Dilations

Example 6: Art ApplicationAn artist is creating a large painting from a photograph by dividing the photograph into squares and dilating each square by a scale factor of 4. If the photograph is 20 cm by 25 cm, what is the perimeter of the painting?

Holt Geometry

12-7 Dilations

Example 7

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.

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 8: 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.

Holt Geometry

12-7 Dilations

Example 8 Continued

It should look like this:

P’

Q’R’

P

R

Q

Holt Geometry

12-7 Dilations

Example 9 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 .

Holt Geometry

12-7 Dilations

Example 9 Continued

It should look like this

RS

TU

R’

S’T’ U’

Holt Geometry

12-7 Dilations

Check Your Understanding

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

Holt Geometry

12-7 Dilations

2. 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

3. 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.