Geometry

Question Analysis

The main focus is applying the property that a perpendicular from the center of a circle to a chord bisects the chord. Then, we'll use the Pythagorean theorem in the right - triangle formed by the radius, the perpendicular, and half of the chord to find .

 

Key Concept Explanation

When a line from the center of a circle is perpendicular to a chord, it divides the chord into two equal parts. In the resulting right - triangle, 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 () to solve for the unknown side.

 

Step - by - Step Solution

1. Let the radius  and the perpendicular distance from the center to the chord .

By the Pythagorean theorem, the length of half of the chord

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