Question

triangle abc is reflected across the y-axis and then dilated by a factor of \\( \frac{1}{2} \\) centered at the origin. which statement correctly describes the resulting image, triangle def?
a. both the reflection and dilation preserve the side lengths and angles of triangle abc.
b. the dilation preserves the side lengths and angles of triangle abc. the reflection does not preserve side lengths and angles.
c. neither the reflection nor the dilation preserves the side lengths and angles of triangle abc. the dilation preserves angles but not side lengths.
d. the reflection preserves the side lengths and angles of triangle abc. the dilation preserves angles but not side lengths.

Explanation:

Step1: Recall Transformations

Reflection is a rigid transformation (isometry), so it preserves side lengths and angles. Dilation with a scale factor (here \( \frac{1}{2} \)) changes side lengths (scales them by the factor) but preserves angles (since it's a similarity transformation).

Step2: Analyze Each Option

• Option A: Dilation does not preserve side lengths, so A is wrong.
• Option B: Reflection preserves side lengths/angles; dilation does not preserve side lengths. So B is wrong.
• Option C: Reflection (rigid) preserves side lengths/angles. Dilation (similarity) preserves angles but not side lengths. So C is wrong.
• Option D: Reflection (isometry) preserves side lengths and angles. Dilation (scale factor \( \frac{1}{2} \)) preserves angles (since it's a similarity transformation) but not side lengths (scales them). So D matches.

Answer:

D. The reflection preserves the side lengths and angles of triangle ABC. The dilation preserves angles but not side lengths.