To determine which composition of similarity transformations maps polygon [tex]\( ABCD \)[/tex] to polygon [tex]\( A'B'C'D' \)[/tex], let's analyze the transformations step-by-step using the given options.
1. Dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a rotation:
- Dilation reduces the size of the polygon by a factor of [tex]\(\frac{1}{4}\)[/tex].
- Following this, a rotation would change the orientation without altering the size again.
2. Dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation:
- Dilation reduces the size of the polygon by a factor of [tex]\(\frac{1}{4}\)[/tex].
- Following this, a translation would move the polygon to a different position without altering its size or shape further.
3. Dilation with a scale factor of 4 and then a rotation:
- Dilation enlarges the size of the polygon by a factor of 4.
- Following this, a rotation would change the orientation without altering the size again.
4. Dilation with a scale factor of 4 and then a translation:
- Dilation enlarges the size of the polygon by a factor of 4.
- Following this, a translation would move the polygon to a different position without altering its size or shape further.
For the polygons [tex]\( ABCD \)[/tex] and [tex]\( A'B'C'D' \)[/tex] to match exactly, let's consider the transformation effects closely.
To map [tex]\( ABCD \)[/tex] to [tex]\( A'B'C'D' \)[/tex] correctly, we need:
- A reduction in size (since a scale factor less than 1 indicates reduction, [tex]\(\frac{1}{4}\)[/tex] indicates reduction).
- A movement to position the resized polygon correctly (translation).
Given these observations, the correct sequence of transformations to achieve the mapping is:
- First, dilate the polygon [tex]\( ABCD \)[/tex] by a scale factor of [tex]\(\frac{1}{4}\)[/tex] to reduce its size.
- Then, translate the reduced polygon to the correct position to match [tex]\( A'B'C'D' \)[/tex].
Therefore, the correct answer is:
A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
Hence, the composition of similarity transformations that maps polygon [tex]\( ABCD \)[/tex] to polygon [tex]\( A'B'C'D' \)[/tex] is indeed:
A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
The answer is: 2.
