In Fig. 10.101, PQ and PR are two tangents to a circle with centre O. If ∠QPR = 46°, then ∠QOR equals

Question

In Fig. 10.101, PQ and PR are two tangents to a circle with centre O. If ∠ QPR = 46 ° , then ∠ QOR equals

Philoid · Accepted Answer

Given:

_∠QPR = 46°

Property 1: The tangent at a point on a circle is at right angles to the radius obtained by joining center and the point of tangency.

Property 2: Sum of all angles of a quadrilateral = 360°.

By property 1, ∆OQP is right-angled at ∠OQP (i.e., ∠OQP = 90°) and ∆ORP is right-angled at ∠ORP (i.e., ∠ORP = 90°).

Now by property 2,

∠OQP + ∠ORP + ∠QOR + ∠QPR = 360°

⇒ ∠QOR = 360° - (∠OQP + ∠ORP + ∠QPR)

⇒ ∠ROP = 360° - (90° + 90° + 46°)

⇒ ∠ROP = 360° - 226°

⇒ ∠ROP = 134°

Hence, ∠ROP = 134°