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:

First, observe the transformation from \(ABCD\) to \(A'B'C'D'\) (not shown in full details but from the coordinate - plane, assume a translation). Then, from \(A'B'C'D'\) to \(A''B'C''D''\), check the rotation rules. A \(180^{\circ}\) rotation about a point \((x_0,y_0)\) of a point \((x,y)\) gives the new point \((2x_0 - x,2y_0 - y)\). A \(270^{\circ}\) rotation about a point \((x_0,y_0)\) of a point \((x,y)\) gives \((y_0+(x - x_0),x_0-(y - y_0))\). By visual inspection (assuming the pre - image \(ABCD\) is translated first and then rotated). A translation is a slide (no rotation or reflection). After an initial translation (to get to \(A'B'C'D'\)), the transformation from \(A'B'C'D'\) to \(A''B'C''D''\) is a rotation. If we consider the orientation of the figure, a \(180^{\circ}\) rotation about \(B'\) (by checking the relative positions of the vertices with respect to \(B'\)) is more consistent with the final image.

Answer:

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