The angle ACB is inscribed in a circle. Point O is the center of the circle. Chord AB = m, angle ACB = x / 2.

The angle ACB is inscribed in a circle. Point O is the center of the circle. Chord AB = m, angle ACB = x / 2. Find the radius of the circle.

1. The degree measure of the inscribed angle is equal to half of the degree measure of the arc on which it rests. Then:
angle ACB = arc AB / 2;
arc AB = 2 * angle ACB;
arc AB = 2 * x / 2 = x.
2. The degree measure of the central angle is equal to the degree measure of the arc on which it rests. The central angle AOB rests on the arc AB, then the angle AOB = x.
3. The chord length is:
l = 2R * sin (α / 2),
where l is the length of the chord, R is the radius of the circle, α is the central angle that rests on the given chord.
The length of the chord AB is equal to m, the central angle that rests on the chord AB is the angle AOB = x. Then:
2R * sin (x / 2) = m;
R = m / 2 * sin (x / 2) (in proportion).
Answer: R = m / 2 * sin (x / 2).