To determine which composition of similarity transformations maps polygon [tex]\( A B C D \)[/tex] to polygon [tex]\( A' B' C' D' \)[/tex], we must analyze each given option step by step.
1. 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 reduce the size of the polygon [tex]\( A B C D \)[/tex] by a factor of 4.
- Following this, a rotation would change the orientation of the polygon without altering its size.
2. 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 reduce the size of the polygon [tex]\( A B C D \)[/tex] by a factor of 4.
- Then, a translation will shift the position of the polygon [tex]\( A B C D \)[/tex] to a new location without changing its size or orientation.
3. Dilation with a scale factor of 4 and then a rotation:
- A dilation with a scale factor of 4 will enlarge the polygon [tex]\( A B C D \)[/tex] by a factor of 4.
- Following this, a rotation would change the orientation of the enlarged polygon without altering its size.
4. Dilation with a scale factor of 4 and then a translation:
- A dilation with a scale factor of 4 will enlarge the polygon [tex]\( A B C D \)[/tex] by a factor of 4.
- Then, a translation will shift the larger polygon [tex]\( A B C D \)[/tex] to a new location without changing its size or orientation.
From the options given, the correct transformation is:
- A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
Therefore, the correct answer is:
- A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
Hence, the option that correctly describes the composition of similarity transformations that maps polygon [tex]\( A B C D \)[/tex] to polygon [tex]\( A' B' C' D' \)[/tex] is:
- [tex]\(\boxed{2}\)[/tex]
