Geometry

Question Analysis

This question pertains to the Perimeters of Similar Shapes. The central task is to leverage the property that for similar triangles, the ratio of their perimeters mirrors the ratio of their corresponding side lengths to determine the required ratio.

Key Concept Explanation

For similar triangles, if the scale factor (the ratio of corresponding side lengths) is , then the ratio of their perimeters is also . This is due to the fact that perimeter is a linear measurement, and similar triangles preserve a consistent proportionality for all linear dimensions.

Step - by - Step Solution

1. First, identify corresponding sides. Let's consider the side of length  in the left triangle and the side of length  in the right triangle (or the side of length  in the left triangle and the side of length  in the right triangle).

2. Calculate the ratio of the corresponding side lengths. Using the sides