Question
Pentagon ABCDE is similar to Pentagon VWXYZ.
ABCDE's perimeter is 36 m. VWXYZ's perimeter is 18 m.
WHich sequence of transformations can be applied to ABCDE to produce VWXYZ?
a diiation with a scale factor of 2 followed by a rotation of 90 degrees clockwise
a translation to the right followed by a dilation with a scale factor of 1/2
a reflection over a vertical line followed by a dilation with a scale factor of 1/2
a rotation of 180 degrees counterclockwise followed by a dilation with a scale factor of 2
ABCDE's perimeter is 36 m. VWXYZ's perimeter is 18 m.
WHich sequence of transformations can be applied to ABCDE to produce VWXYZ?
a diiation with a scale factor of 2 followed by a rotation of 90 degrees clockwise
a translation to the right followed by a dilation with a scale factor of 1/2
a reflection over a vertical line followed by a dilation with a scale factor of 1/2
a rotation of 180 degrees counterclockwise followed by a dilation with a scale factor of 2
Answers
GPT-4o mini
To transform Pentagon ABCDE into Pentagon VWXYZ while maintaining their similarity, we need to consider the scale factor between their perimeters.
The perimeter of ABCDE is 36 m and the perimeter of VWXYZ is 18 m. To find the scale factor, we can divide the perimeter of ABCDE by the perimeter of VWXYZ:
\[
\text{Scale Factor} = \frac{\text{Perimeter of ABCDE}}{\text{Perimeter of VWXYZ}} = \frac{36}{18} = 2
\]
Since VWXYZ is smaller than ABCDE, the appropriate scale factor for the transformation will be \(\frac{1}{2}\) (indicating that VWXYZ is half the size of ABCDE, not bigger).
Now let's analyze the given options for transformations:
1. **A dilation with a scale factor of 2 followed by a rotation of 90 degrees clockwise:** This will not produce Pentagon VWXYZ because the dilation is incorrect (it enlarges rather than shrinks).
2. **A translation to the right followed by a dilation with a scale factor of 1/2:** While the dilation is correct (it will reduce the size of the pentagon), the translation may not be necessary depending on the position of the pentagons.
3. **A reflection over a vertical line followed by a dilation with a scale factor of 1/2:** This is a valid sequence since the reflection can reposition the pentagon, and the dilation will properly shrink it down to the appropriate size.
4. **A rotation of 180 degrees counterclockwise followed by a dilation with a scale factor of 2:** Similar to the first option, the dilation is incorrect (it enlarges rather than reduces).
Based on the understanding from the options provided, the best choice is:
**A reflection over a vertical line followed by a dilation with a scale factor of 1/2.**
The perimeter of ABCDE is 36 m and the perimeter of VWXYZ is 18 m. To find the scale factor, we can divide the perimeter of ABCDE by the perimeter of VWXYZ:
\[
\text{Scale Factor} = \frac{\text{Perimeter of ABCDE}}{\text{Perimeter of VWXYZ}} = \frac{36}{18} = 2
\]
Since VWXYZ is smaller than ABCDE, the appropriate scale factor for the transformation will be \(\frac{1}{2}\) (indicating that VWXYZ is half the size of ABCDE, not bigger).
Now let's analyze the given options for transformations:
1. **A dilation with a scale factor of 2 followed by a rotation of 90 degrees clockwise:** This will not produce Pentagon VWXYZ because the dilation is incorrect (it enlarges rather than shrinks).
2. **A translation to the right followed by a dilation with a scale factor of 1/2:** While the dilation is correct (it will reduce the size of the pentagon), the translation may not be necessary depending on the position of the pentagons.
3. **A reflection over a vertical line followed by a dilation with a scale factor of 1/2:** This is a valid sequence since the reflection can reposition the pentagon, and the dilation will properly shrink it down to the appropriate size.
4. **A rotation of 180 degrees counterclockwise followed by a dilation with a scale factor of 2:** Similar to the first option, the dilation is incorrect (it enlarges rather than reduces).
Based on the understanding from the options provided, the best choice is:
**A reflection over a vertical line followed by a dilation with a scale factor of 1/2.**