Question

Question 5
If a line segment joining midpoints of two chords of a circle passes through the centre of the circle, prove that the two chords are parallel.

Answer 1

Given AB and CD are two chords of a circle whose centre is O and PQ is a diameter bisecting the chord AB and CD at L and M, respectively and the diameter PQ passes through the centre O of the circle.

To prove that AB || CD

Proof
L is the mid-point of AB as OP is perpendicular bisector of AB.
$\therefore OL\perp AB$
[since, the line joining the centre of a circle to the mid-point of a chord is perpendicular to the chord]
$\Rightarrow \angle ALO={90}^{\circ }$....(i)
$Similarly,OM\perp CD$
$\therefore \angle OMD={90}^{\circ }$....(ii)
From Eqs. (i) and (ii), $\angle ALO=\angle OMD={90}^{\circ }$
But, these are alternate angles
So AB || CD, hence proved.