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To determine which composition of similarity transformations can map polygon [tex]\( A_{BCD} \)[/tex] to polygon [tex]\( A'B'C'D' \)[/tex], let's carefully consider the steps involved in similarity transformations. These include dilation (scaling), translation (shifting), and rotation (turning).
Given the options:
1. A dilation with a scale factor of [tex]\( \frac{1}{4} \)[/tex] and then a rotation.
2. A dilation with a scale factor of [tex]\( \frac{1}{4} \)[/tex] and then a translation.
3. A dilation with a scale factor of 4 and then a rotation.
4. A dilation with a scale factor of 4 and then a translation.
Here is a logical step-by-step solution:
1. Dilation with a Scale Factor of [tex]\( \frac{1}{4} \)[/tex]:
- Dilation means resizing the polygon. If we use a scale factor of [tex]\( \frac{1}{4} \)[/tex], each side of polygon [tex]\( A_{BCD} \)[/tex] will become [tex]\( \frac{1}{4} \)[/tex] of its original length. The shape remains similar, but the size is reduced by a factor of 4.
2. Translation:
- After the polygon is resized (dilated), a translation moves every point of the polygon the same distance in a given direction. This step is necessary when the final positions of the vertices of the polygon [tex]\( A'B'C'D' \)[/tex] are at different locations compared to those of [tex]\( A_{BCD} \)[/tex], but still preserving their relative positions.
Considering the options provided:
- Option 1 (A dilation with a scale factor of [tex]\( \frac{1}{4} \)[/tex] and then a rotation): This transformation involves resizing the polygon and then rotating it. While this could change the polygon's orientation, it does not guarantee the final positions of the vertices as required.
- Option 2 (A dilation with a scale factor of [tex]\( \frac{1}{4} \)[/tex] and then a translation): This involves resizing and then shifting the entire polygon to the required position. This option effectively changes both the size and the location without altering the orientation of the polygon.
- Option 3 (A dilation with a scale factor of 4 and then a rotation): This involves enlarging the polygon by a factor of 4, which is the opposite of what the task requires, followed by a rotation. This is not suitable as the resulting polygon will be much larger than [tex]\( A'B'C'D' \)[/tex].
- Option 4 (A dilation with a scale factor of 4 and then a translation): Similar to option 3, this involves enlarging the polygon, which does not align with reducing the size to map [tex]\( A_{BCD} \)[/tex] to [tex]\( A'B'C'D' \)[/tex].
The correct answer is:
- A dilation with a scale factor of [tex]\( \frac{1}{4} \)[/tex] and then a translation.
Thus, the composition of similarity transformations that maps polygon [tex]\( A_{BCD} \)[/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.
Given the options:
1. A dilation with a scale factor of [tex]\( \frac{1}{4} \)[/tex] and then a rotation.
2. A dilation with a scale factor of [tex]\( \frac{1}{4} \)[/tex] and then a translation.
3. A dilation with a scale factor of 4 and then a rotation.
4. A dilation with a scale factor of 4 and then a translation.
Here is a logical step-by-step solution:
1. Dilation with a Scale Factor of [tex]\( \frac{1}{4} \)[/tex]:
- Dilation means resizing the polygon. If we use a scale factor of [tex]\( \frac{1}{4} \)[/tex], each side of polygon [tex]\( A_{BCD} \)[/tex] will become [tex]\( \frac{1}{4} \)[/tex] of its original length. The shape remains similar, but the size is reduced by a factor of 4.
2. Translation:
- After the polygon is resized (dilated), a translation moves every point of the polygon the same distance in a given direction. This step is necessary when the final positions of the vertices of the polygon [tex]\( A'B'C'D' \)[/tex] are at different locations compared to those of [tex]\( A_{BCD} \)[/tex], but still preserving their relative positions.
Considering the options provided:
- Option 1 (A dilation with a scale factor of [tex]\( \frac{1}{4} \)[/tex] and then a rotation): This transformation involves resizing the polygon and then rotating it. While this could change the polygon's orientation, it does not guarantee the final positions of the vertices as required.
- Option 2 (A dilation with a scale factor of [tex]\( \frac{1}{4} \)[/tex] and then a translation): This involves resizing and then shifting the entire polygon to the required position. This option effectively changes both the size and the location without altering the orientation of the polygon.
- Option 3 (A dilation with a scale factor of 4 and then a rotation): This involves enlarging the polygon by a factor of 4, which is the opposite of what the task requires, followed by a rotation. This is not suitable as the resulting polygon will be much larger than [tex]\( A'B'C'D' \)[/tex].
- Option 4 (A dilation with a scale factor of 4 and then a translation): Similar to option 3, this involves enlarging the polygon, which does not align with reducing the size to map [tex]\( A_{BCD} \)[/tex] to [tex]\( A'B'C'D' \)[/tex].
The correct answer is:
- A dilation with a scale factor of [tex]\( \frac{1}{4} \)[/tex] and then a translation.
Thus, the composition of similarity transformations that maps polygon [tex]\( A_{BCD} \)[/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.
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