Question

In the given figure, TBP and TCQ are tangents to the circle whose centre is O. Also, ∠PBA = 60° and ∠ACQ =70°. Determine ∠BAC and ∠BТС.

Answer 1

Given:

TBP and TCQ are tangents at B and C, respectively.

∠PBA = 60° and ∠ACQ = 70°.

1. Using Tangent–Chord Theorem

Angle between a tangent and a chord equals the angle in the opposite arc.

(a) At point B:

∠PBA = ∠BCA

∠BCA = 60°

(b) At point C:

∠ACQ = ∠CBA

∠CBA = 70°

2. Finding ∠BAC

In triangle ABC:

∠BAC = 180° − (∠BCA + ∠CBA)

= 180° − (60° + 70°)

= 50°

3. Finding ∠BTC

For two tangents drawn from an external point T:

∠BTC = 180° − ∠BOC

(where ∠BOC is the central angle subtending arc BC)

Arc BC subtends:

∠BAC = 50° = half of ∠BOC

Therefore,

∠BOC = 2 × 50° = 100°

Thus,

∠BTC = 180° − 100°

= 80°