Geometry

Question Analysis

The main focus is on using the relationship between a chord, the perpendicular from the center to the chord, and the radius of the circle. We'll apply the Pythagorean theorem to find the value of  (half of the chord length).

 

Key Concept Explanation

When a line from the center of a circle is perpendicular to a chord, it bisects the chord. In the right - triangle formed, the radius is the hypotenuse, the perpendicular distance from the center to the chord is one leg, and half of the chord length is the other leg. We use the Pythagorean theorem  (where  is the hypotenuse and ,  are the legs) to solve for the unknown side.

 

Step - by - Step Solution

1. Here, the radius of the circle  and the perpendicular distance from the center to the chord . Let  be half of the chord length (the other leg of the right - triangle).

2. Apply the Pythagorean theorem: .

First, calculate

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