To determine which composition of similarity transformations maps polygon \(ABCD\) to polygon \(A'B'C'D'\), we need to carefully assess the given transformations and understand their effects.
1. Dilation with a Scale Factor of \(\frac{1}{4}\):
- Dilation is a transformation that scales a polygon by a given factor relative to a fixed point, commonly the origin.
- A scale factor of \(\frac{1}{4}\) reduces the size of the polygon. Each side of polygon \(ABCD\) becomes \(\frac{1}{4}\) times its original length. This will significantly shrink the polygon.
2. Translation:
- Translation is a transformation that shifts all points of a polygon by a certain distance in a specified direction without altering its shape or size.
3. Rotation:
- Rotation is a transformation that turns the polygon around a fixed point, typically the origin, by a certain angle.
Given these transformations:
- A dilation with a scale factor of \( \frac{1}{4} \) results in a smaller, similar polygon.
- After the dilation, if we follow it with a translation, we effectively move the smaller polygon to a new position without changing its size or shape further.
This combination correctly transforms \(ABCD\) to \(A'B'C'D'\):
Thus, the correct choice of similarity transformations is:
- A dilation with a scale factor of \(\frac{1}{4}\) and then a translation.
Therefore, the composition of similarity transformations that maps polygon \(ABCD\) to polygon \(A'B'C'D'\) is:
- a dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
