Question

Prove that the segment joining the points of contact of a two parallel tangent passes through the centre.

Answer 1

To prove: $AOB$ is a straight line passing through $O$.

Let $PAQ$ and $RBS$ be two parallel tangents to a circle with centre $O$.

Join $OA$ and $OB$. Draw $OC||PQ$.

Now, $PA||CO$

$\Rightarrow$ $\angle PAO+\angle COA={180}^o$ [Sum of the angle on the same side of a transversal is ${180}^o$]

$\Rightarrow$ ${90}^o+\angle COA={180}^o$ [$\because \angle PAQ=$angle between a tangent and radius=${90}^o$]

$\Rightarrow$ $\angle COA={90}^o$

Similarly, $\angle COB={90}^o$

$\therefore \angle COA+\angle COB={90}^o+{90}^o={180}^o$

Hence, $AOB$ is a straight line passing through $O$.