To determine the correct statement about the transformations performed on triangle ABC, let's analyze each transformation step-by-step:
1. Reflection across the [tex]$y$[/tex]-axis:
- A reflection is a type of rigid transformation, meaning it preserves the size and shape of the figure.
- Therefore, after reflecting triangle ABC across the [tex]$y$[/tex]-axis, triangle DEF will have the same side lengths and angles as triangle ABC, but its orientation will be reversed.
2. Dilation by a factor of [tex]$\frac{1}{2}$[/tex] centered at the origin:
- A dilation changes the size of the figure but keeps the shape the same.
- Specifically, a dilation with a factor of [tex]$\frac{1}{2}$[/tex] reduces the length of each side of triangle DEF to half of the corresponding side length in triangle ABC.
- However, the angles of the triangle are preserved during a dilation.
Given these properties, we can now evaluate each statement:
A. The reflection preserves the side lengths and angles of triangle ABC. The dilation preserves angles but not side lengths.
- This statement is correct. Reflection preserves both side lengths and angles, while dilation preserves angles but reduces side lengths.
B. Both the reflection and dilation preserve the side lengths and angles of triangle ABC.
- This statement is incorrect because the dilation does not preserve side lengths; it changes them by a factor of [tex]$\frac{1}{2}$[/tex].
C. Neither the reflection nor the dilation preserves the side lengths and angles of triangle ABC.
- This statement is incorrect because the reflection does preserve both side lengths and angles. The dilation preserves angles but not side lengths.
D. The dilation preserves the side lengths and angles of triangle ABC. The reflection does not preserve side lengths and angles.
- This statement is incorrect because the dilation does not preserve side lengths; it only preserves angles. Additionally, the reflection does preserve both side lengths and angles.
Thus, the correct answer is:
A. The reflection preserves the side lengths and angles of triangle ABC. The dilation preserves angles but not side lengths.
