To determine which composition of similarity transformations maps polygon [tex]\( A B C D \)[/tex] to polygon [tex]\( A' B' C' D' \)[/tex], we need to understand the given options and their effects on the original polygon.
### Similarity Transformations Recap:
1. Dilation: This transformation scales the polygon by a certain factor, either enlarging or reducing it.
2. Rotation: This transformation turns the polygon around a fixed point.
3. Translation: This transformation shifts the polygon in the plane without altering its shape or size.
Given the options, we need to evaluate each combination:
1. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a rotation
- The polygon will be reduced in size by a factor of [tex]\(\frac{1}{4}\)[/tex] (i.e., shrunk to one-quarter its original size).
- Then, this smaller polygon will be rotated.
2. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation
- The polygon will be reduced in size by a factor of [tex]\(\frac{1}{4}\)[/tex].
- The smaller polygon will then be shifted to a different location.
3. A dilation with a scale factor of 4 and then a rotation
- The polygon will be enlarged by a factor of 4 (i.e., each side will become four times longer).
- This larger polygon will then be rotated.
4. A dilation with a scale factor of 4 and then a translation
- The polygon will be enlarged by a factor of 4.
- The larger polygon will then be shifted to a different location.
To match polygon [tex]\( A B C D \)[/tex] to polygon [tex]\( A' B' C' D' \)[/tex]:
We need the original polygon to be scaled by 4 times first and then translated to align with the position and orientation of the target polygon. The combination of these transformations ensures that the final polygon has the correct size and position.
### Conclusion:
The correct composition of similarity transformations that maps polygon [tex]\( A B C D \)[/tex] to polygon [tex]\( A' B' C' D' \)[/tex] is a dilation with a scale factor of 4 and then a translation.
Thus, the answer is:
A dilation with a scale factor of 4 and then a translation.
