Question

In the following figure, ‘O’ is the centre of the circle. If ∠ABO = 20° and ∠ACO = 30°, find ∠BOC.

Answer 1

Given:

O is the centre of the circle

∠ABO = 20°

∠ACO = 30°

We need to find ∠BOC (the central angle subtended by arc BC)

Step 1: Use triangle ABO

OB = OA (radii of the circle), so it is isosceles.

Therefore, base angles at A and B are equal.

∠ABO = 20°,

∠BAO = 20°

∠AOB = 180° − (20° + 20°) = 140°

Step 2: Use triangle ACO

OC = OA (radii), so AC = AC and it is isosceles.

So, base angles are equal.

∠ACO = 30°,

∠CAO = 30°

∠AOC = 180° − (30° + 30°) = 120°

Step 3: Find ∠BOC

∠BOC = ∠BOA + ∠AOC

= 140° + 120°

= 260°

But angles around a point = 360°, and ∠BOC is the angle between OB and OC inside the circle.

∠BOC = 360° − 260°

= 100°