Question

In Fig.  APB is a tangent to a circle with centre O at point P. If ∠QPB = 500, then the measure of ∠POQ is

A. 100° 

B. 120° 

C. 140° 

D. 150°

Answer 1

Answer is A. 100°

Given: 

∠QPB =50° 

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 triangle = 180°. 

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

∠OPQ = ∠OPB – ∠QPB 

⇒ ∠OPQ = 90° – 50° = 40° 

And, 

∠OPQ = ∠OQP [∵ OP = OQ (radius of circle)] 

Now by property 2, 

∠OPQ + ∠OQP + ∠POQ = 180° 

⇒ 40° + 40° + ∠POQ = 180° 

⇒ 80° + ∠POQ = 180° 

⇒ ∠POQ = 180° - 80° 

⇒ ∠POQ = 100°

Hence, ∠POQ = 100°