Question

If a line segment joining mid-points 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: AB || CD

Proof: Since, L is the mid-point of AB.

∴ OL ⊥ AB  ...[Since, the line joining the centre of a circle to the mid-point of a chord is perpendicular to the chord]

⇒ ∠ALO = 90°   ...(i)

Similarly, OM ⊥ CD

∴ ∠OMD = 90°  ...(ii)

From equations (i) and (ii),

∠ALO = ∠OMD = 90° 

But, these are alternating angles.

So, AB || CD

Hence proved.