Inscribed and Central Angles in Circles

Definitions and theorems related to inscribed and central angles in circles are discussed using examples.

Definitions

1. A central angle of a circle is an angle whose vertex is at the center of the circle. For example, \(\angle BOC\) in the figure below.

2. An inscribed angle is an angle whose vertex is on the circle and whose sides are chords. For example, \(\angle CAB\) in the figure below.

Theorems

1. The measure of an inscribed angle is half the measure of the central angle intercepting the same arc: \[ \angle BAC = \frac{1}{2} \angle BOC \] 2. Two or more inscribed angles intercepting the same arc are congruent: \[ \angle BAC = \angle BDC \]

Problem

In the figure below, chord \(CA = 12 \text{ cm}\). The circle with center \(O\) has a radius of \(14 \text{ cm}\). Find the measure of the inscribed angle \(\angle CBA\) (approximate to two decimal places).

Solution

1. First, find the central angle \(\angle COA\). Triangle \(COA\) is isosceles because \(CO = AO = 14 \text{ cm}\) (radii).
2. Use the Law of Cosines in \(\triangle COA\): \[ CA^2 = CO^2 + AO^2 - 2(CO)(AO)\cos(\angle COA) \] Substitute known values: \[ 12^2 = 14^2 + 14^2 - 2(14)(14)\cos(\angle COA) \] \[ 144 = 392 - 392\cos(\angle COA) \] \[ \cos(\angle COA) = \frac{392 - 144}{392} = \frac{248}{392} = \frac{31}{49} \]
3. Therefore: \[ \angle COA = \arccos\left(\frac{31}{49}\right) \]
4. By the inscribed angle theorem: \[ \angle CBA = \frac{1}{2} \angle COA = \frac{1}{2} \arccos\left(\frac{31}{49}\right) \] Evaluating: \[ \angle CBA \approx 25.38^\circ \]

More Geometry Tutorials and Problems