Answer :
Let's analyze the transformations mentioned in the question to determine which composition maps polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex].
We have two composition options:
1. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
2. A dilation with a scale factor of 4 and then a translation.
### Understanding Dilation and Translation
#### 1. Dilation
A dilation is a transformation that changes the size of a figure but not its shape. It scales the figure by a certain factor:
- Dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex]:
- This will reduce the size of polygon [tex]\(ABCD\)[/tex] to one-quarter of its original size.
- Dilation with a scale factor of 4:
- This would enlarge the size of polygon [tex]\(ABCD\)[/tex] to four times its original size.
#### 2. Translation
Translation is a transformation that shifts a figure from one location to another without changing its size or shape. The specific details of the translation are not provided, but the key information is that the figure is moved to a different location.
### Step-by-Step Solution
Given that the transformation involves mapping [tex]\(ABCD\)[/tex] to [tex]\(A'B'C'D'\)[/tex]:
1. Dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation:
- First, you reduce the size of polygon [tex]\(ABCD\)[/tex] so that each side is one-quarter of its original length.
- Then, you shift this smaller polygon to a new location to create [tex]\(A'B'C'D'\)[/tex].
2. Dilation with a scale factor of 4 and then a translation:
- First, you enlarge the size of polygon [tex]\(ABCD\)[/tex] so that each side is four times its original length.
- Then, you shift this enlarged polygon to a new location.
Given that the desired composition should map polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex] correctly, the appropriate transformation should involve reducing the size of polygon [tex]\(ABCD\)[/tex] first (since the scale factor [tex]\(\frac{1}{4}\)[/tex] is mentioned) and then translating it.
Thus, the correct composition of similarity 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.
We have two composition options:
1. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
2. A dilation with a scale factor of 4 and then a translation.
### Understanding Dilation and Translation
#### 1. Dilation
A dilation is a transformation that changes the size of a figure but not its shape. It scales the figure by a certain factor:
- Dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex]:
- This will reduce the size of polygon [tex]\(ABCD\)[/tex] to one-quarter of its original size.
- Dilation with a scale factor of 4:
- This would enlarge the size of polygon [tex]\(ABCD\)[/tex] to four times its original size.
#### 2. Translation
Translation is a transformation that shifts a figure from one location to another without changing its size or shape. The specific details of the translation are not provided, but the key information is that the figure is moved to a different location.
### Step-by-Step Solution
Given that the transformation involves mapping [tex]\(ABCD\)[/tex] to [tex]\(A'B'C'D'\)[/tex]:
1. Dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation:
- First, you reduce the size of polygon [tex]\(ABCD\)[/tex] so that each side is one-quarter of its original length.
- Then, you shift this smaller polygon to a new location to create [tex]\(A'B'C'D'\)[/tex].
2. Dilation with a scale factor of 4 and then a translation:
- First, you enlarge the size of polygon [tex]\(ABCD\)[/tex] so that each side is four times its original length.
- Then, you shift this enlarged polygon to a new location.
Given that the desired composition should map polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex] correctly, the appropriate transformation should involve reducing the size of polygon [tex]\(ABCD\)[/tex] first (since the scale factor [tex]\(\frac{1}{4}\)[/tex] is mentioned) and then translating it.
Thus, the correct composition of similarity 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.
