Question

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.

Answer 1

(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 = $\frac{1}{2}\times 60°$ = 30º

Thus, the measure of ∠ACB is 30º.

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

Thus, the measure of arc AB is 30º.

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

Thus, the measure of arc ACB is 300º.