Geometry

Question Analysis

The main focus is on using the Pythagorean theorem to find the distance from the center of the circle to the chord, and then determining the value of  based on the radius and that distance.

 

Key Concept Explanation

In a circle, when a perpendicular is drawn from the center to a chord, a right - triangle is formed with the radius as the hypotenuse, half of the chord length as one leg, and the distance from the center to the chord as the other leg. The Pythagorean theorem () is used to relate these sides.

 

Step - by - Step Solution

1. First, find the distance from the center of the circle to the chord using the Pythagorean theorem.

Let the radius  and half of the chord length . Let the distance from the center to the chord be .

According to the Pythagorean theorem .

Substitute  and  into the formula:

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