In the given figure, in a circle with centre O, length of chord AB is equal to the radius of the circle. Find measure of each of the following.

(1) ∠ AOB (2)∠ ACB

(3) arc AB (4) arc ACB.

#### Solution

(1) In the given figure, OA and OB are the radii of the circle.

OA = OB = AB (Given)

∴ ∆OAB is an equilateral triangle.

⇒ ∠AOB = ∠OAB = ∠OBA = 60º

Thus, the measure of ∠AOB is 60º.

(2) The measure of an angle subtended by an arc at a point on the circle is half of the measure of the angle subtended by the arc at the centre.

∠ACB = \[\frac{1}{2}\] ∠AOB = \[\frac{1}{2}\] x 60° = 30°

Thus, the measure of ∠ACB is 30º.

(3) m(arc AB) = ∠AOB = 60º (Measure of an arc is the measure of its corresponding central angle)

Thus, the measure of arc AB is 60º.

(4) m(arc ACB) = 360º − m(arc AB) = 360º − 60º = 300º

Thus, the measure of arc ACB is 300º.