To determine which composition of similarity transformations maps polygon [tex]\( ABCD \)[/tex] to polygon [tex]\( A'B'C'D' \)[/tex], let's go through the possible choices and reason our way to the correct answer.
1. Choice 1: A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a rotation
- First, a dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] (which makes the polygon smaller by a factor of 4).
- Then, a rotation around some point.
2. Choice 2: A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation
- First, a dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex], reducing the size of the polygon.
- Then, a translation, which shifts the scaled polygon to a new position without changing its size or orientation.
3. Choice 3: A dilation with a scale factor of 4 and then a rotation
- First, a dilation with a scale factor of 4, enlarging the polygon by a factor of 4.
- Then, a rotation around some point.
4. Choice 4: A dilation with a scale factor of 4 and then a translation
- First, a dilation with a scale factor of 4, enlarging the polygon.
- Then, a translation, shifting the enlarged polygon.
Analyzing the transformations needed to map polygon [tex]\( ABCD \)[/tex] to polygon [tex]\( A'B'C'D' \)[/tex]:
- A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] reduces the polygon's size.
- After scaling down the polygon, the most logical step to map it appropriately to the location of [tex]\( A'B'C'D' \)[/tex] would be translating it to match the final position.
Based on this logical reasoning, the correct composition of similarity transformations that maps polygon [tex]\( ABCD \)[/tex] to polygon [tex]\( A'B'C'D' \)[/tex] is:
Choice 2: A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
