In Fig. 10.96, if TP and TQ are tangents drawn from an external point T to a circle with centre O such that ∠TQP = 60°, then ∠OPQ =

Question

In Fig. 10.96, if TP and TQ are tangents drawn from an external point T to a circle with centre O such that ∠ TQP = 60 ° , then ∠ OPQ =

Philoid · Accepted Answer

Given:

_∠TQP = 60°

Property 1: If two tangents are drawn to a circle from one external point, then their tangent segments (lines joining the external point and the points of tangency on circle) are equal.

Property 2: The tangent at a point on a circle is at right angles to the radius obtained by joining center and the point of tangency.

By property 1,

TP = TQ (tangent from T)

⇒ ∠TPQ = ∠TQP = 60°

By property 2, ∆OPT is right-angled at ∠OPT (i.e., ∠OPT = 90°) and ∆OQT is right-angled at ∠OQT (i.e., ∠OQT = 90°).

Now,

∠OPQ = ∠OPT – ∠TPQ

⇒ ∠OPQ = 90° – 60°

⇒ ∠OPQ = 30°

Hence, ∠OPQ = 30°