Question

In the given figure, PQ = QR and ∠RQP = 72°. CP and CQ are tangents. Determine ∠POQ.

Answer 1

Given:

PQ = QR

∠RQP = 72°

CP and CQ are tangents to the circle.

O is the centre of the circle.

To find: ∠POQ.

Step 1: Use the triangle property

Since PQ = QR, triangle PQR is isosceles.

So, base angles are equal:

∠RPQ = ∠PRQ

Let each angle = x.

We know in triangle PQR:

x + x + 72° = 180°

2x = 108°

x = 54°

Thus,

∠RPQ = 54°

∠PRQ = 54°

Step 2: Use the tangent–radius theorem

The radius drawn to the point of tangency is perpendicular to the tangent.

So:

OQ ⟂ CQ

OP ⟂ CP

Thus, angles between PQ and OQ, and between PR and OP, relate directly to the triangle angles.

Relation between ∠PRQ and arc PQ

Step 3: Relation between inscribed and central angles.

Angle POQ is the central angle subtending arc PQ.

Angle PRQ (54°) is the inscribed angle subtending the same arc PQ.

Central angle = 2 × Inscribed angle

∠POQ = 2 × ∠PRQ

∠POQ = 2 × 54°

∠POQ = 108°