**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.**