Question

what is the final transformation in the composition of transformations that maps pre - image abcd to image abcd?
a translation down and to the right
a translation up and to the right
a 270° rotation about point b
a 180° rotation about point b

Explanation:

1. First, analyze the transformation from \(A'B'C'D'\) to \(A''B''C''D''\). The pre - image \(A'B'C'D'\) and the center of rotation \(B'\) are given.
2. A \(180^{\circ}\) rotation about a point \(B'\) will map a point \((x,y)\) relative to \(B'\) as \((-x,-y)\) relative to \(B'\). Looking at the figure, the transformation from \(A'B'C'D'\) to \(A''B''C''D''\) shows the characteristics of a \(180^{\circ}\) rotation about \(B'\) (the angles and side - length relationships match a \(180^{\circ}\) rotation, while a \(270^{\circ}\) rotation would have a different orientation, and translations up/down and right don't match the rotational - like change from \(A'B'C'D'\) to \(A''B''C''D''\)).

Answer:

D. a \(180^{\circ}\) rotation about point \(B'\)