Question

Prove that the line joining the midpoints of two parallel chords of a circle passes through the centre.

Answer 1

Let $AB$ and $CD$ be two parallel chords having $P$ and $Q$ as their mid-points respectively. Let $O$ be the centre of the circle. Join $OP$ and $OQ$ and draw $OX\parallel AB$ or $CD$.

Now, $P$ is the mid-point of $AB$

$OP\perp AB,\angle BPO={90}^{\circ }$

But, $OX\parallel AB$.

Therefore, $\angle XOQ=\angle BPO$ [Corresponding angles]

$\angle XOQ={90}^{\circ }$

Similarly, $Q$ is the mid-point of $CD$

$OQ\perp CD,\angle DQO={90}^{\circ }$

But, $OX\parallel CD$.

Therefore, $\angle POX=\angle DQO={90}^{\circ }$ [Corresponding angles]

$\angle POX+\angle XOQ={90}^{\circ }+{90}^{\circ }={180}^{\circ }$

$POQ$ is a straight line.

Hence, $PQ$ is a straight line passing through the centre of the circle.