Question

In the given diagram, O is the centre of the circle. PR and PT are two tangents drawn from the external point P and touching the circle at Q and S respectively. MN is a diameter of the circle. Given ∠PQM = 42° and ∠PSM = 25°.

Find:

1. ∠OQM
2. ∠QNS
3. ∠QOS
4. ∠QMS

Answer 1

a. PR and PT are tangents to the circle with centre O.

Then, ∠OQP = 90°

As radius is ⊥ to the tangent

Then, ∠OQM = ∠OQP – ∠MQP

= 90° – 42°

= 48°

b. ∠PQM = ∠QNM = 42°  ...(By alternate segment theorem)

∠PSM = ∠SNM = 25°

Then ∠QNS = ∠QNM + ∠SNM

= 42° + 25°

= 67°

c. ∠QOS = 2∠QNS   ...(since the angle subtended by the arc at the centre is twice the angle subtended by the arc at any other point of the circle.)

= 2 × 67°

= 134°

d. QNSN is a cyclic quadrilateral

∠QNS + ∠QMS = 180°

∠QMS = 180° – 67°

= 113°