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 given options systematically.
### Options
1. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a rotation:
- Dilation: If we apply a dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex], the size of the polygon [tex]\(ABCD\)[/tex] will shrink to [tex]\(\frac{1}{4}\)[/tex] of its original size.
- Rotation: After shrinking, if we apply a rotation, the new polygon would be the same size but rotated. This doesn't account for translation, only a change in orientation and size.
2. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation:
- Dilation: When we apply the dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex], the polygon [tex]\(ABCD\)[/tex] will again shrink to [tex]\(\frac{1}{4}\)[/tex] of its original size.
- Translation: Following the dilation, if we then apply a translation (a shift in position without altering the shape or size), we can place the resized polygon in the correct location to align with [tex]\(A'B'C'D'\)[/tex].
3. A dilation with a scale factor of 4 and then a rotation:
- Dilation: Applying a dilation with a scale factor of 4 will enlarge the polygon [tex]\(ABCD\)[/tex] to four times its original size.
- Rotation: After enlarging, a rotation will only change the orientation and not address any positional shift needed for alignment with [tex]\(A'B'C'D'\)[/tex].
4. A dilation with a scale factor of 4 and then a translation:
- Dilation: With a scale factor of 4, the polygon [tex]\(ABCD\)[/tex] will expand to four times its initial size.
- Translation: A subsequent translation will relocate the enlarged polygon, but this composition doesn’t align with the typically observed need for reduction in size to meet a smaller target polygon, as outlined in the problem description.
### Conclusion
Given the options, the transformation that effectively adjusts the size correctly to [tex]\(\frac{1}{4}\)[/tex] and subsequently places the resized polygon in the desired position accurately without altering orientation is:
A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
Thus, the correct composition of similarity transformations to map polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex] is:
[tex]\(\boxed{2}\)[/tex]
