Geometry

Question Analysis

The main focus is to apply the Secant - Tangent Rule. We know the lengths of part of the secant segment and , and is expressed in terms of . We need to find the value of first and then the length of .

 

Key Concept Explanation

Secant - Tangent Rule: If a tangent line and a secant line are drawn from an external point to a circle, with the tangent length , the external part of the secant , and the entire secant length , then .

If a tangent line and a secant line are drawn from an external point to a circle, then .

 

Step - by - Step Solution

1. Set up the equation using the rule:

We know that , , so , and .

According to the Secant - Tangent Rule, .

2. Solve the equation for :

First, calculate , so

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