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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 :

GinnyAnsweridn
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.
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