Question

1. (from unit 3, lesson 6.) find a sequence of rigid transformations and dilations that takes square abcd to square efgh.

a. translate by the directed line segment ae, which will take b to a point b. then rotate with center e by angle bef. finally, dilate with center e by scale factor \\(\frac{5}{2}\\).
b. translate by the directed line segment ae, which will take b to a point b. then rotate with center e by angle bef. finally, dilate with center e by scale factor \\(\frac{2}{5}\\).
c. dilate using center e by scale factor \\(\frac{2}{5}\\).
d. dilate using center e by scale factor \\(\frac{5}{2}\\).

Explanation:

Step1: Analyze Side Lengths

Square \(ABCD\) has side length \(5\) (from \(AB = 5\)), square \(EFGH\) has side length \(2\) (from \(EF = 2\)). The scale factor for dilation should be \(\frac{\text{length of } EFGH}{\text{length of } ABCD}=\frac{2}{5}\).

Step2: Analyze Transformations

First, translate \(ABCD\) by \(AE\) to align \(A\) with \(E\) (taking \(B\) to \(B'\)). Then rotate around \(E\) to align the sides (angle \(B'EF\)). Finally, dilate with center \(E\) by scale factor \(\frac{2}{5}\) to match the side lengths. Option a uses scale factor \(\frac{5}{2}\) (wrong, would enlarge), c skips translation/rotation (can't align without them), d is wrong scale. Option b has correct translation, rotation, and scale factor \(\frac{2}{5}\).

Answer:

b. Translate by the directed line segment \(AE\), which will take \(B\) to a point \(B'\). Then rotate with center \(E\) by angle \(B'EF\). Finally, dilate with center \(E\) by scale factor \(\frac{2}{5}\).