Question

what is the final transformation in the composition of transformations that maps pre - image abcd to image a\bc\d\?
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:

To determine the final transformation, we analyze the steps from \( A'B'C'D' \) to \( A''B''C''D'' \). First, identify the position of \( B' \) (from the translated pre - image). The figure \( A'B'C'D' \) is transformed to \( A''B''C''D'' \) by a rotation about \( B' \). A \( 180^\circ \) rotation about a point \( (x,y) \) maps a point \( (x_1,y_1) \) to \( (2x - x_1,2y - y_1) \). By observing the coordinates (or the shape's orientation change), we see that the transformation from \( A'B'C'D' \) to \( A''B''C''D'' \) is a \( 180^\circ \) rotation about \( B' \). The other options (translations or \( 270^\circ \) rotation) do not match the orientation and position change.

Answer:

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