Q) Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contacts at the centre.

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, ∠ QRP  + ∠ POQ  = 180°

From  the diagram, we can see that RQOP forms a quadrilateral and we know that, the sum of all 4 angles in any quadrilateral is 360⁰.

∠ QRP + ∠ RPO + ∠ POQ + ∠ RQO  = 360° ……. (i)

Since we know that a tangent is always perpendicular to the radius, therefore

∠ RPO = ∠ RQO = 90°

Substituting these values in equation (i), we get

∠ QRP + 90° + ∠ POQ + 90°  = 360°

∠ QRP  + ∠ POQ  = 180° …… Hence Proved!