To determine which composition of similarity transformations maps polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex], let's analyze the given transformations and their effects on the polygon.
Let's break down each option provided:
1. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a rotation:
- A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] will shrink the polygon [tex]\(ABCD\)[/tex] to a polygon that is one-fourth the size of the original polygon.
- After this dilation, a rotation will change the orientation of the shrunken polygon, but not its size.
2. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation:
- A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] will shrink the polygon [tex]\(ABCD\)[/tex] to a polygon that is one-fourth the size of the original polygon.
- A translation will then move the entire shrunken polygon to another location without altering its size or orientation.
3. A dilation with a scale factor of 4 and then a rotation:
- A dilation with a scale factor of 4 will enlarge the polygon [tex]\(ABCD\)[/tex] to a polygon that is four times the size of the original polygon.
- After this dilation, a rotation will change the orientation of the enlarged polygon, but not its size.
4. A dilation with a scale factor of 4 and then a translation:
- A dilation with a scale factor of 4 will enlarge the polygon [tex]\(ABCD\)[/tex] to a polygon that is four times the size of the original polygon.
- A translation will then move the entire enlarged polygon to another location without altering its size or orientation.
Given these descriptions and understanding that the correct choice is:
4. A dilation with a scale factor of 4 and then a translation
This combination means that the polygon [tex]\(ABCD\)[/tex] is first enlarged to four times its original size through a dilation. After the polygon is enlarged, it is translated (shifted to another position) while maintaining its shape and orientation, but at the new, larger size.
Hence, the composition of similarity transformations that maps polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex] is a dilation with a scale factor of 4 and then a translation.
