Arcs and Angles
General Explanation
To solve problems involving triangles within circles, like the one shown below:
these facts should be kept in mind:
Sample Problem
A triangle with vertices A, B, and C is inscribed in a circle with center 0
and radius 1. Suppose AB = AC and that Angle BAC = 100°. Find the measure of Angle CBO. Solution
The only angle we are given is Angle BAC. Since it is an inscribed angle, its measure is half that of the arc it cuts off. Arc BC therefore
must have twice the measure of Angle BAC, so arc BC = 200°.
Because there are 360 degrees in a circle, Arc BC + arc BAC = 360°. Since arc BC = 200°, arc BAC must equal 160°. Angle BOC is a central angle, and so must be the measure of the angle it cuts off. Therefore, Angle BOC = arc BAC = 160°. Since BO and CO are both radii, BO = CO. The triangle they form is isosceles, so Angle BCO = Angle CBO. We know one of the angles of triangle BOC and that the other two angles are the same. If we let x be one of the other two angles, then x + x + 160° = 180°. Solving this equation, we get x = 10°, so the answer is 10 degrees. |