1. Similarity and Dilations
Clusters: 1. Understand similarity in terms of similarity transformations 2. Prove theorems involving similarity.

2. 05 Dilations on the Coordinate Plane
Learning Target
I can define dilation.
I can perform a dilation with a given center and scale factor on a figure in the coordinate plane.

3. Connection to previews lesson…
Previously, we studied rigid transformations, in which the image and preimage of a figure are congruent. In this lesson, you will study a type of nonrigid transformation called a dilation, in which the image and preimage of a figure are similar.

4. Dilations
A dilation is a type of transformation that enlarges or reduces a figure but the shape stays the same.
The dilation is described by a scale factor and a center of dilation.

5. Dilations
The scale factor k is the ratio of the length of any side in the image to the length of its corresponding side in the preimage. It describes how much the figure is enlarged or reduced.

6. PQR ~ P´Q´R´, is equal to the scale factor of the dilation. P´Q´ PQ
The dilation is a reduction if k < 1 and it is an enlargement if k > 1. P P´ 6 5 P´ P 3 2 Q´ Q • • R C C R´ Q R´ Q´ R
Reduction: k = = = 3 6 1 2
CP CP
Enlargement: k = = 5 2
CP CP
PQR ~ P´Q´R´, is equal to the scale factor of the
dilation.
P´Q´ PQ

7. Constructing a Dilation
Examples of constructed a dilation of a triangle.

8. Steps in constructing a dilation
Step 1: Construct ABC on a coordinate plane with A(3, 6), B(7, 6), and C(7, 3).

9. Steps in constructing a dilation
Step 2: Draw rays from the origin O through A, B, and C. O is the center of dilation.

10. Steps in constructing a dilation
Step 3: With your compass, measure the distance OA. In other words, put the point of the compass on O and your pencil on A. Transfer this distance twice along OA so that you find point A’ such that OA’ = 3(OA). That is, put your point on A and make a mark on OA. Finally, put your point on the new mark and make one last mark on OA. This is A’.

11. Steps in constructing a dilation

12. Steps in constructing a dilation
Step 4: Repeat Step 3 with points B and C.
That is, use your compass to find points B’ and C’ such that OB’ = 3(OB) and OC’ = 3(OC).

13. Steps in constructing a dilation

14. Steps in constructing a dilation
You have now located three points, A’, B’, and C’, that are each 3 times as far from point O as the original three points of the triangle.
Step 5: Draw triangle A’B’C’.
A’B’C’ is the image of ABC under a dilation with center O and a scale factor of 3. Are these images similar?

15. Steps in constructing a dilation

16. Questions/ Observations:
Step 6: What are the lengths of AB and A’B’? BC and B’C’? What is the scale factor?
AB = 4
A’B’= 12
BC = 3
B’C’= 9

17. Questions/ Observations:
Step 7: Measure the coordinates of A’, B’, and C’.
Image
A´(9, 18)
B´(21, 18)
C´(21, 9)

18. Questions/ Observations:
Step 8: How do they compare to the original coordinates?
P(x, y)  P´(kx, ky)
Pre-image
A(3, 6) 
B(7, 6) 
C(7, 3) 
Image
A´(9, 18)
B´(21, 18)
C´(21, 9)

19. In a coordinate plane, dilations whose centers are the origin have the property that the image of P(x, y) is P´(kx, ky).
Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4).
Use the origin as the center and use a scale factor of How does the perimeter of the preimage compare to the perimeter of the image? 1 2
SOLUTION
Because the center of the dilation is the origin, you can find the image of each vertex by multiplying its coordinates by the scale factor. y D C
A(2, 2)  A´(1, 1) D´ A C´ B A´
B(6, 2)  B´(3, 1) 1 B´ •
C(6, 4)  C ´(3, 2) O 1 x
D(2, 4)  D´(1, 2)

20. A preimage and its image after a dilation are similar figures. •
In a coordinate plane, dilations whose centers are the origin have the property that the image of P(x, y) is P´(kx, ky).
Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4). Use the origin as the center and use a scale factor of How does the perimeter of the preimage compare to the perimeter of the image? 1 2
SOLUTION y
From the graph, you can see that the preimage has a perimeter of 12 and the image has a perimeter of 6. D C D´ A C´ B
A preimage and its image after a dilation are similar figures. A´ 1 B´ • O 1 x
Therefore, the ratio of the perimeters of a preimage and its image is equal to the scale factor of the dilation.

21. Example 3
Determine if ABCD and A’B’C’D’ are similar figures. If so, identify the scale factor of the dilation that maps ABCD onto A’B’C’D’ as well as the center of dilation.
Is this a reduction or an enlargement?