Geometry

Question Analysis

This question focuses on applying trigonometric functions, specifically the tangent function, in the context of similar right - angled triangles. The main task is to find the tangent of ∠Q by leveraging the equality of ∠Q and ∠BAO and using 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, allowing 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 opposite is , and the side adjacent to is .

3. Calculate the tangent value

Using the formula , for

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