Question

Prove that the line segments joining the points of contact of two parallel tangents of a circle,passes through its centre ?

Answer 1

Let, $PQ$ and $RS$ be two parallel tangents to a circle with cantre $O$ where $M,N$ be the points of contact of the two tangents $PQ,RS$ to the circle respectively.

We have to prove that the line $MN$ passes through the centre $O$ of the circle.

Here, $OM$ and $OA$ are joined.

$OA\parallel PQPM\parallel AO\therefore \angle PMO+\angle AOM={180}^o\left[sumoftwoadjacentinteriorangles\right]$

but, $\angle PMO={90}^o[\because Atangenttoacircleisperpendiculartotheradiusthroughpointofcontact]$

So, ${90}^o+\angle AOM={180}^o\Rightarrow \angle AOM={90}^o\therefore \angle AON={90}^o\angle AOM+\angle AON={180}^o$

$\therefore$ $MN$ is a straight line and it passes through $O$ which is the centre of the given circle.

Hence, proved.