Question

In the following figure, PQ and PR are tangents to the circle, with centre O. If `∠`QPR = 60°, calculate:

1. ∠QOR,
2. ∠OQR,
3. ∠QSR.

Answer 1

Join QR.

i. In quadrilateral ORPQ,

OQ ⊥ OP, OR ⊥ RP

∴ ∠OQP = 90°, ∠ORP = 90°, ∠QPR = 60°

∠QOR = 360° – (90° + 90° + 60°)

∠QOR = 360° – 240°

∠QOR = 120°

ii. In ΔQOR,

OQ = QR  ...(Radii of the same circle)

∴ ∠OQR = ∠QRO  ...(i)

But, ∠OQR + ∠QRO + ∠QOR = 180°

∠OQR + ∠ QRO + 120° = 180°

∠OQR + ∠QRO = 60°

From (i)

2∠OQR = 60°

∠OQR = 30°

iii. Now arc RQ subtends ∠QOR at the centre and ∠QSR at the remaining part of the circle.

∴ `∠QSR = 1/2 ∠QOR`

`=> ∠QSR = 1/2 xx 120^circ`

`=>` ∠QSR = 60°