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. 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. the reflection preserves the side lengths and angles of triangle abc. the dilation preserves angles but not side lengths.
d. neither the reflection nor the dilation preserves the side lengths and angles of triangle abc.

Explanation:

Step1: Recall Reflection Properties

Reflection is a rigid transformation. Rigid transformations (like reflection, rotation, translation) preserve side lengths and angles (they produce congruent figures). So reflection of triangle \(ABC\) will preserve its side lengths and angles.

Step2: Recall Dilation Properties

Dilation is a non - rigid transformation. When we dilate a figure with a scale factor \(k\) (here \(k = \frac{1}{2}\)), the side lengths of the image are \(k\) times the side lengths of the original figure. However, the angles of the figure remain the same as the original figure (since dilation is a similarity transformation, it preserves angles and produces similar figures). So dilation preserves angles but not side lengths.

Step3: Analyze Each Option

• Option A: Dilation does not preserve side lengths, so this is incorrect.
• Option B: Reflection preserves side lengths and angles, dilation preserves angles but not side lengths. So this statement is incorrect.
• Option C: Reflection (rigid transformation) preserves side lengths and angles. Dilation (non - rigid, similarity transformation) preserves angles but not side lengths. This statement is correct.
• Option D: Reflection does preserve side lengths and angles, so this is incorrect.

Answer:

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