Question

Polygon ABCD is dilated, rotated, and translated to form polygon A′B′C′D′. The endpoints of AB are at (0, -7) and (8, 8), and the endpoints of AB are at (6, -6) and (2, 1.5). What is the scale factor of the dilation?

Answer 1

Rotation and translation are rigid transformations, they don't change figure sizes. Dilation change figure sizes increasing or decreasing them by scale factor.

First, find AB and A'B' by the formula:

$AB=\sqrt{(x_B-x_A)^2+(y_B-y_A)^2}= \sqrt{(8-0)^2+(8-(-7))^2}=\sqrt{8^2+15^2}=\sqrt{64+225}=\sqrt{289}=17,\\ \\A'B'=\sqrt{(x_{B'}-x_{A'})^2+(y_{B'}-y_{A'})^2}= \sqrt{(2-6)^2+(1.5-(-6))^2}=\sqrt{4^2+7.5^2}=\sqrt{16+56.25}=\sqrt{72.25}=8.5.$

As you can see AB=2A'B'. This means that the segment AB was decreased twice to form segment A'B'. Then the scale factor is 1/2.