Question

In the adjoining figure 'O' is the center of the circle and PQ, PR and ST are the three tangents. ∠QPR = 50°, then ∠SOT equals 

(a) 35° 

(b) 65° 

(c) 45° 

(d) 50°

Answer 1

Answer : (b) 65º 

∠ROQ = 180° – 50° = 130° (∵ ∠OQP + ∠ORP + ∠OPR + ∠ROQ = 360° and ∠OQP = ∠ORP = 90°) 

RT = TM, QS = SM (Tangents to a circle from the same external point are equal) 

Also, OQ = OM = OR (Radii of the given circle) 

∴  ∠ROT = ∠TOM and ∠MOS = ∠SOQ. (∵ Tangents from the an external point subtend equal angles at the centre)

⇒ ∠SOT = ∠SOM + ∠TOM = \(\frac{1}{2}\) ∠QOM + \(\frac{1}{2}\) ∠ROM 

∠SOT = \(\frac{1}{2}\) ∠ROQ 

= \(\frac{1}{2}\) × 130° 

= 65°.