Question

If tangents PA and PB from a point P to a circle with centre ‘O’ are inclined to each other at angle of 80°, then find the measured of ∠POA.

Answer 1

Between ΔAOP and ΔBOP 

∠OAP = ∠OBP = 90° (∵ tangent and line from origin meet at 90° to each other) 

OA=OB = radius of the circle = r (say) and OP is the common side 

Hence, we can say ΔAOP = ΔBOP 

Therefore, we can say 

∠OPA = ∠OPB = \(\frac{1}{2}\)(∠APB) 

= \(\frac{1}{2}\)(80°) = 40° (∵ given that tangents PA and PB are inclined to each other by 80°) 

Now, in ΔAOP,

∠POA + ∠OPA + ∠A = 180° 

⇒ ∠POA + 40° + 90° = 180° 

⇒ ∠POA = 50°