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 use the side - length ratios of the given right - triangle to find the tangent of ∠O.

 

Key Concept Explanation

In a right - triangle, for an acute angle , .

 

Step - by - Step Solution

1. In right - triangle  (where ), we know  and .

Since  and  are similar (by AA similarity or other geometric relationships in the figure), the ratios of corresponding sides are the same.

For , the side opposite  is  (corresponding to ) and the side adjacent to  is  (corresponding to ).

Because of similarity, the ratio of the opposite side to the adjacent side for  is the same as that in .

Using the formula