Question

In the given figure, O is the centre of the circle. Tangents at A and B meet at C. If ∠ACO = 30°, find:

1. ∠BCO 
2. ∠AOB
3. ∠APB

Answer 1

In the given fig, O is the centre of the circle and CA and CB are the tangents to the circle from C. Also, ∠ACO = 30°

P is any point on the circle. PA and PB are joined.

To find:  

1. ∠BCO 
2. ∠AOB
3. ∠APB

Proof:

i. In ΔOAC and OBC

OC = OC  ...(Common)

OA = OB   ...(Radius of the circle)

CA = CB   ...(Tangents to the circle)

∴  ΔOAC ≅  ΔOBC  ...(SSS congruence criterion)

∴ ∠ACO = ∠BCO = 30°

ii. ∠ACB = 30° + 30° = 60°

∴ ∠AOB + ∠ACB = 180°

`=>` ∠AOB + 60° = 180°

`=>` ∠AOB = 180° – 60°

`=>` ∠AOB = 120°

iii. Arc AB subtends ∠AOB at the centre and ∠APB is in the remaining part of the circle.

∴ `∠APB = 1/2 ∠AOB`

= `1/2 xx 120^circ`

= 60°