Similarity, Right Triangles, and Trigonometry Clusters: 1. Understand similarity in terms of similarity transformations 2. Prove theorems involving similarity. 3. Define trigonometric ratios and solve problems involving right triangles. 4. Apply trigonometry to general triangles.

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.

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.

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.

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.

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

Constructing a Dilation Examples of a dilation of a triangle.

Questions/ Observations: 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)

Example 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?