Question

select the correct answer.
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. the reflection preserves the side lengths and angles of triangle abc. the dilation preserves angles but not side lengths.
b. both the reflection and dilation preserve the side lengths and angles of triangle abc.
c. neither the reflection nor the dilation preserves the side lengths and angles of triangle abc.
d. the dilation preserves the side lengths and angles of triangle abc. the reflection does not preserve side lengths and angles.

Explanation:

1. Recall the properties of reflection and dilation:

• Reflection: A reflection is a rigid transformation. Rigid transformations (like reflection, translation, rotation) preserve both the side lengths and the angles of a figure. So, reflecting triangle \(ABC\) across the \(y\)-axis will result in a congruent triangle (same side lengths and angles).
• Dilation: A dilation is a non - rigid transformation. When we dilate a figure by a scale factor \(k=\frac{1}{2}\) centered at the origin, the side lengths of the figure are multiplied by the scale factor \(k\). However, the angles of the figure remain the same because dilation is a similarity transformation (it preserves the shape, so angles are congruent).

1. Analyze each option:

• Option A: The reflection preserves side lengths and angles (correct for reflection). The dilation preserves angles but not side lengths (since side lengths are scaled by \(\frac{1}{2}\)). This option correctly describes the properties of reflection and dilation.
• Option B: Dilation does not preserve side lengths, so this option is incorrect.
• Option C: Reflection is a rigid transformation that preserves side lengths and angles, so this option is incorrect.
• Option D: Reflection preserves side lengths and angles, and dilation preserves angles, so this option is incorrect.

Answer:

A. The reflection preserves the side lengths and angles of triangle \(ABC\). The dilation preserves angles but not side lengths.