When we transform a shape through reflection and dilation, it's important to understand how these transformations affect various properties of the shape such as side lengths and angles.
First, consider the reflection across the [tex]\( y \)[/tex]-axis:
1. Reflection Across the [tex]\( y \)[/tex]-Axis:
- When a triangle is reflected across the [tex]\( y \)[/tex]-axis, its image will be a mirror image.
- The reflection preserves both the side lengths and angles of the original triangle. This is because the reflection is an isometry, meaning the distances and angles are maintained.
- Therefore, after reflecting Triangle ABC across the [tex]\( y \)[/tex]-axis, the resulting triangle still has the same side lengths and angles as Triangle ABC.
Next, consider the dilation centered at the origin by a factor of [tex]\(\frac{1}{2}\)[/tex]:
2. Dilation by a Factor of [tex]\(\frac{1}{2}\)[/tex] Centered at the Origin:
- Dilation is a transformation that changes the size of a figure but not its shape.
- Specifically, a dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] will reduce the side lengths of the triangle to half of their original lengths.
- This dilation maintains the same angles because all linear dimensions are scaled by the same factor. Thus, the proportional relationships between the side lengths remain, and the angles stay unchanged.
- Therefore, after the dilation, Triangle DEF will have side lengths that are half of those of the reflected triangle, but the angles will remain the same.
Putting these two transformations together:
- The reflection across the [tex]\( y \)[/tex]-axis preserves the side lengths and angles of Triangle ABC.
- The dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] preserves the angles but scales the side lengths by [tex]\(\frac{1}{2}\)[/tex].
Therefore, the correct statement describing the resulting image, triangle DEF, would be:
A. The reflection preserves the side lengths and angles of triangle ABC. The dilation preserves angles but not side lengths.
