Answer :
To determine which statement correctly describes the resulting image, triangle [tex]\( \triangle DEF \)[/tex], after reflecting [tex]\( \triangle ABC \)[/tex] across the [tex]\( y \)[/tex]-axis and then dilating by a factor of [tex]\(\frac{1}{2}\)[/tex] centered at the origin, we need to analyze the effects of each transformation.
### Reflection Across the [tex]\( y \)[/tex]-axis:
1. Side lengths: A reflection across the [tex]\( y \)[/tex]-axis changes the orientation of the shape but does not alter the side lengths. Therefore, side lengths are preserved.
2. Angles: Reflections are rigid transformations, meaning they do not change the angles within the shape. Thus, angles are preserved as well.
### Dilation by a Factor of [tex]\(\frac{1}{2}\)[/tex]:
1. Side lengths: A dilation changes the size of the shape based on the given factor. In this case, a dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] will reduce all side lengths of the original triangle [tex]\( ABC \)[/tex] to half of their original lengths. Thus, side lengths are not preserved.
2. Angles: Dilations centered at a point preserve the angles within the shape. In other words, the relative proportions and angles within the shape remain the same, even though the size changes.
Taking these analyses into account:
- Reflection across the [tex]\( y \)[/tex]-axis preserves both side lengths and angles of [tex]\( \triangle ABC \)[/tex].
- Dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] preserves angles but not side lengths.
Therefore, the correct statement is:
D. The reflection preserves the side lengths and angles of triangle [tex]\( ABC \)[/tex]. The dilation preserves angles but not side lengths.
### Reflection Across the [tex]\( y \)[/tex]-axis:
1. Side lengths: A reflection across the [tex]\( y \)[/tex]-axis changes the orientation of the shape but does not alter the side lengths. Therefore, side lengths are preserved.
2. Angles: Reflections are rigid transformations, meaning they do not change the angles within the shape. Thus, angles are preserved as well.
### Dilation by a Factor of [tex]\(\frac{1}{2}\)[/tex]:
1. Side lengths: A dilation changes the size of the shape based on the given factor. In this case, a dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] will reduce all side lengths of the original triangle [tex]\( ABC \)[/tex] to half of their original lengths. Thus, side lengths are not preserved.
2. Angles: Dilations centered at a point preserve the angles within the shape. In other words, the relative proportions and angles within the shape remain the same, even though the size changes.
Taking these analyses into account:
- Reflection across the [tex]\( y \)[/tex]-axis preserves both side lengths and angles of [tex]\( \triangle ABC \)[/tex].
- Dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] preserves angles but not side lengths.
Therefore, the correct statement is:
D. The reflection preserves the side lengths and angles of triangle [tex]\( ABC \)[/tex]. The dilation preserves angles but not side lengths.
