Q) From an external point, two tangents are drawn to a circle. Prove that the line joining the external point to the centre of the circle bisects the angle between the two tangents.

Ans:

Let’s draw a diagram with a circle of radius r and O as centre. Let the two tangents RP & RQ are drawn from point R on to this circle, The tangent RP touches the circle at point P and tangent RQ touches the circle at point Q.

We need to prove that, line OR bisects ∠ QRP and to do that we need to prove that ∠ QRO = ∠ PRO

From  the diagram, we can see that:

OQ = OP (radii of same circle)

∠ RPO = ∠ RQO = 90° (tangent is to the radius)

RP = RQ   (tangents from same point to a circle are always same)

   Δ ROP Δ ROQ

Hence, ∠ QRO = ∠ PRO

Therefore it is proved that the line OR bisects ∠ QRP.