6.7: Similarity Transformations Objectives: 1.To use dilations to create similar figures 2.To perform dilations in the coordinate plane using coordinate notation

Example 1 Use the coordinate mapping ( x, y ) → ( x + 8, y + 3) to translate  SAM to create  S’A’M’.

Dilations dilation A dilation is a type of transformation that enlarges or reduces a figure. scale factorcenter of dilation 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.

Investigation 1 Complete the following investigation to construct a dilation of a triangle.

Investigation 1 Step 1: Construct ΔABC on a coordinate plane with A(3, 6), B(7, 6), and C(7, 3).

Investigation 1 Step 2: Draw rays from the origin O through A, B, and C. O is the center of dilation.

Investigation 1 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’.

Investigation 1 Step 3:

Investigation 1 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).

Investigation 1 Step 4:

Investigation 1 Step 5: 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. Draw triangle A’B’C’. ΔA’B’C’ is the image of ABC under a dilation with center O and a scale factor of 3. Are these images similar?

Investigation 1 Step 5:

Investigation 1 Step 6: What are the lengths of AB and A’B’? BC and B’C’? What is the scale factor?

Investigation 1 Step 7: Measure the coordinates of A’, B’, and C’.

Investigation 1 Step 8: How do they compare to the original coordinates?

Example 2 What happens to any point ( x, y ) under a dilation centered at the origin with a scale factor of k ?

Dilations in the Coordinate Plane You can describe a dilation with respect to the origin with the notation ( x, y ) → ( kx, ky ), where k is the scale factor.

Dilations in the Coordinate Plane You can describe a dilation with respect to the origin with the notation ( x, y ) → ( kx, ky ), where k is the scale factor. Enlargement: Enlargement: k > 1.

Dilations in the Coordinate Plane You can describe a dilation with respect to the origin with the notation ( x, y ) → ( kx, ky ), where k is the scale factor. Reduction: Reduction: 0 < k < 1.

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?

Example 4 A graph shows  PQR with vertices P(2, 4), Q(8, 6), and R(6, 2), and segment ST with endpoints S(5, 10) and T(15, 5). At what coordinate would vertex U be placed to create ΔSUT, a triangle similar to ΔPQR?

Example 5 Figure J’K’L’M’N’ is a dilation of figure JKLMN. Find the coordinates of J’ and M’.