To determine which composition of similarity transformations maps polygon [tex]\( ABCD \)[/tex] to polygon [tex]\( A'B'C'D' \)[/tex], we should evaluate the options given:
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] would reduce the size of the polygon to one-fourth of its original size.
- Following the dilation, performing a rotation would rotate the already reduced-sized polygon by a certain angle around a fixed point. This does not involve any shifting of the position other than the rotational shift.
2. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation:
- Similarly, a dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] would reduce the size of the polygon.
- Following the dilation, performing a translation would move (or shift) the entire reduced-sized polygon to a different location without changing its orientation.
3. A dilation with a scale factor of 4 and then a rotation:
- A dilation with a scale factor of 4 would increase the size of the polygon to four times its original size.
- Following the dilation, performing a rotation rotates the larger polygon around a fixed point. This also does not involve any shifting of the position other than the rotational shift.
4. A dilation with a scale factor of 4 and then a translation:
- A dilation with a scale factor of 4 would increase the size of the polygon to four times its original size.
- Following the dilation, performing a translation shifts (or moves) the entire larger polygon to a different location without changing its orientation.
Given that the correct composition of transformations is a dilation with a scale factor of 4 and then a translation, it means we are looking at option 4:
- The polygon [tex]\(ABCD\)[/tex] is first enlarged by a factor of 4 to become 4 times its original size.
- Then, the newly resized polygon is translated, which means moved to a different position without altering its orientation.
Thus, the appropriate 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.
