Geometry

Question Analysis

This question focuses on applying trigonometric functions, specifically the cosine function, in the context of similar right - angled triangles. The main task is to find the cosine of ∠X by using the equality of ∠X and ∠ZEF and the side - length ratios of the relevant right - triangle.

 

Key Concept Explanation

In a right - triangle, for an acute angle , . When two triangles are similar, corresponding angles are equal, enabling us to use the side - length relationships of one triangle for the trigonometric calculation of the corresponding angle in the other triangle.

 

Step - by - Step Solution

1. Identify the equal angles and relevant triangle

Since (by the properties of similar triangles or geometric relationships in the figure), we can find by calculating .

2. Determine the sides for calculating

In right - triangle , . The side adjacent to is , and the hypotenuse is .

3. Calculate the cosine value

Using the formula , for

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