Geometry

Question Analysis

The main focus of this question is to complete the proportion using the properties of similar triangles. It requires identifying the similar triangles in the diagram and applying the concept of equal ratios of corresponding sides. This tests understanding of geometric similarity and the ability to correctly match corresponding sides in similar triangles.

 

Key Concept Explanation

If two triangles are similar, the ratios of their corresponding side lengths are equal. In this case, since  is common to both  and , and  (by the angle markings),  by the AA (angle - angle) similarity criterion. For similar triangles, if we have side lengths  in one triangle and  in the other, then .

 

Step - by - Step Solution

1. Identify the similar triangles:

Given that  is common and , we can conclude that  by the AA similarity criterion.

2. Set up the proportion for corresponding sides:

We know that  is one ratio of corresponding sides. The side

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