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Which composition of similarity transformations maps polygon [tex]$ABCD$[/tex] to polygon [tex]$A^{\prime}B^{\prime}C^{\prime}D^{\prime}$[/tex]?

A. A dilation with a scale factor of [tex]$\frac{1}{4}$[/tex] and then a rotation

B. A dilation with a scale factor of [tex]$\frac{1}{4}$[/tex] and then a translation

C. A dilation with a scale factor of 4 and then a rotation

D. A dilation with a scale factor of 4 and then a translation


Sagot :

To determine which composition of similarity transformations maps polygon [tex]\( ABCD \)[/tex] to polygon [tex]\( A'B'C'D' \)[/tex], let's go through the possible choices and reason our way to the correct answer.

1. Choice 1: A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a rotation

- First, a dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] (which makes the polygon smaller by a factor of 4).
- Then, a rotation around some point.

2. Choice 2: A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation

- First, a dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex], reducing the size of the polygon.
- Then, a translation, which shifts the scaled polygon to a new position without changing its size or orientation.

3. Choice 3: A dilation with a scale factor of 4 and then a rotation

- First, a dilation with a scale factor of 4, enlarging the polygon by a factor of 4.
- Then, a rotation around some point.

4. Choice 4: A dilation with a scale factor of 4 and then a translation

- First, a dilation with a scale factor of 4, enlarging the polygon.
- Then, a translation, shifting the enlarged polygon.

Analyzing the transformations needed to map polygon [tex]\( ABCD \)[/tex] to polygon [tex]\( A'B'C'D' \)[/tex]:

- A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] reduces the polygon's size.
- After scaling down the polygon, the most logical step to map it appropriately to the location of [tex]\( A'B'C'D' \)[/tex] would be translating it to match the final position.

Based on this logical reasoning, the correct composition of similarity transformations that maps polygon [tex]\( ABCD \)[/tex] to polygon [tex]\( A'B'C'D' \)[/tex] is:

Choice 2: A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
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