Question

If PQ is a tangent to the circle at R; calculate:

1. ∠PRS,
2. ∠ROT.

Given O is the centre of the circle and angle TRQ = 30°.

Answer 1

PQ is a tangent and OR is the radius.

∴ OR ⊥ PQ

∴ ∠ORT = 90°

`=>` ∠TRQ = 90° – 30° = 60°

But in ΔOTR,

OT = OR  ...(Radii of the same circle)

∴ ∠OTR = 60° Or ∠STR = 60°

But,

∠PRS = ∠STR = 60  ...(Angle in the alternate segment)

In ΔORT,

∠ORT = 60°

∠OTR = 60°

∴ ∠ROT = 180° – (60° + 60°) 

∴ ∠ROT = 180° – 120° = 60°