To determine which composition of similarity transformations maps polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex], we need to consider the possible effects of each transformation on the polygon. We are provided with four options for the transformations:
1. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a rotation
2. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation
3. A dilation with a scale factor of 4 and then a rotation
4. A dilation with a scale factor of 4 and then a translation
Let's analyze each option step by step:
1. Dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a rotation:
- Dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] would shrink the polygon [tex]\(ABCD\)[/tex] to [tex]\(\frac{1}{4}\)[/tex] of its original size.
- Following this, a rotation would change the orientation of the polygon but not its size.
- This transformation does change the size and orientation of the polygon.
2. Dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation:
- Dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] would shrink the polygon [tex]\(ABCD\)[/tex] to [tex]\(\frac{1}{4}\)[/tex] of its original size.
- Following this, a translation would shift the position of the polygon without changing its size or orientation.
- This transformation would result in a polygon of the appropriate size in a new position but with the same orientation as before.
3. Dilation with a scale factor of 4 and then a rotation:
- Dilation with a scale factor of 4 would enlarge the polygon [tex]\(ABCD\)[/tex] to 4 times its original size.
- Following this, a rotation would change the orientation of the polygon but not its size.
- This transformation does not match the requirement implied in the problem.
4. Dilation with a scale factor of 4 and then a translation:
- Dilation with a scale factor of 4 would enlarge the polygon [tex]\(ABCD\)[/tex] to 4 times its original size.
- Following this, a translation would shift the position of the polygon without changing its size or orientation.
- This also does not match the requirement implied in the problem.
Given these analyses, the composition of similarity transformations that correctly maps polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex] is:
A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
