1. Objective
Identify and draw dilations.

2. 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.

3. Example 1: Identifying Dilations
Tell whether each transformation appears to be a dilation. Explain. A. B.
No; the figures are not similar.
Yes; the figures are similar and the image is not turned or flipped.

4. Check It Out! Example 1
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.

5. 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

7. A dilation enlarges or reduces all dimensions proportionally
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.

8. 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. W’ X’
Step 3 Connect the vertices of the image. Y’ Z’

9. Step 1 Draw a line through Q and each vertex.
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. R’ S’
Step 2 On each line, mark twice the distance from Q to the vertex.
Step 3 Connect the vertices of the image. T’ U’

10. 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.

11. 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

13. 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.

14. 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

15. Graph the preimage and image.
Example 4 Continued
Graph the preimage and image. P P’ Q’ R’ R Q

16. 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

17. Check It Out! Example 4 Continued
Graph the preimage and image. R’ S’ T’ U’ R S T U

18. Practice Quiz
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 .