Question

In the given figure, O is the centre of the circle. Tangents at A and B meet at C. If ∠ACO = 30°,
find: (i) ∠ BCO (ii) ∠ AOB (iii) ∠ APB

Answer 1

(i) ∠ BCO = ∠ ACO = 30°   .....( ∴ C is the intersecting point of tangents AC and BC)

(ii) ∠ OAC = ∠ OBC = 90°

∠ ACO = 30°                      .....(Given)

∠ AOC = ∠ BOC = 180° - (90° + 30°)  ....(Sum of the angles of a Δ is 180°)

∠ AOC = 180° - 120°

∠ AOC = 60°

∠ AOB = ∠ AOC + ∠ BOC

∠ AOB = 60° + 60° = 120°

(iii) ∠ APB = ${\displaystyle \frac{1}{2}\text{∠ AOB}=\frac{120°}{2}=60°}$    .....( ∴ Angle substended at the remaining part of the circle is half the ∠ substended at the centre)