1. 12-7
Dilations
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry

2. 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.

3. Objective
TSW identify and draw dilations.

4. Vocabulary
center of dilation
enlargement
reduction

5. 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.

6. Example 1: Identifying Dilations
Tell whether each transformation appears to be a dilation. Explain. A. B.

7. Example 2
Tell whether each transformation appears to be a dilation. Explain. a. b.

8. 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

10. 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.

11. 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.

12. Example 3: Drawing Dilations
You should get something that looks like this: W’ X’ Y’ Z’

13. 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.

14. Example 4 R’ S’ T’ U’

15. 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.

16. Example 6: Art Application
An 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?

17. 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.

19. 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.

20. 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.

21. It should look like this:
Example 8 Continued
It should look like this: P P’ Q’ R’ R Q

22. 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

23. It should look like this
Example 9 Continued
It should look like this R’ S’ T’ U’ R S T U

24. Check Your Understanding
1. Tell whether the transformation appears to be a dilation.

25. Lesson Quiz: Part II
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.
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.