Question

In the given figure, PQ is chord of a circle with centre O an PT is a tangent. If
∠QPT = 60°, find the ∠PRQ.

 

Answer 1

We know that the radius and tangent are perpendicular at their point of contact
∴ ∠OPT = 90°
Now,∠OPQ = ∠OPT - ∠QPT = 90° - 60° = 30°
Since, OP = OQas born is radius
∴ ∠OPQ  = ∠OQP = 30° (Angles opposite to equal sides are equal)
Now, In isosceles, POQ
∠POQ + ∠OPQ +∠OQP = 180° (Angle sum property of a triangle)
⇒ ∠POQ =180° - 30° - 30° =120°
Now, ∠POQ + reflex ∠POQ = 360° (Complete angle)
⇒ reflex  ∠POQ = 360° - 120°  = 240°
We know that the angle subtended by an arc at the centre double the angle subtended by the arc at any point on the remaining part of the circle
∴ ∠PRQ `=1/2 `( reflex ∠POQ)= 120°