Geometry

Question Analysis

The main focus is on using the perpendicular from the center of the circle to the chord property along with the Pythagorean theorem. We'll find the value of  (radius) by considering the right - triangle formed by the radius, the perpendicular from the center to the chord, and half of the chord.

 

Key Concept Explanation

When a line from the center of a circle is perpendicular to a chord, it bisects the chord. In the resulting right - triangle, the radius of the circle 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 can use the Pythagorean theorem  (where  is the hypotenuse and ,  are the legs) to solve for the radius.

 

Step - by - Step Solution

1. Since the perpendicular from the center to the chord bisects the chord, half of the chord length is . The perpendicular distance from the center to the chord is .

2. Apply the Pythagorean theorem:

.

Calculate