To determine which composition of similarity transformations maps polygon [tex]\( ABCD \)[/tex] to polygon [tex]\( A'B'C'D' \)[/tex], let's analyze the possible combinations of transformations.
Firstly, let's understand the types of transformations being discussed:
1. Dilation: This transformation changes the size of the polygon while maintaining the shape. The scale factor tells us by how much the size changes. A scale factor of [tex]\(\frac{1}{4}\)[/tex] means the polygon's size is reduced to one-fourth of its original size. A scale factor of 4 means the polygon's size is increased to four times its original size.
2. Translation: This transformation moves the polygon from one location to another without changing its shape or size.
3. Rotation: This transformation rotates the polygon around a fixed point but does not change its size.
Given the options:
- "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] and then a translation"
- "a dilation with a scale factor of 4 and then a rotation"
- "a dilation with a scale factor of 4 and then a translation"
From these options, we already understand that the correct composition involves reducing the polygon by a factor of [tex]\(\frac{1}{4}\)[/tex].
Next, we consider the second transformation. We need to determine whether it is a translation or a rotation. Upon reflection and proper reasoning, we conclude that the correct combination in this context is reducing the size of the polygon and then translating it to its final position.
Therefore, the correct sequence of transformations that maps polygon [tex]\( ABCD \)[/tex] to polygon [tex]\( A'B'C'D' \)[/tex] is:
A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
