Question

The line joining the mid-points of two chords of a circle passes through its centre. Prove that the chords are parallel.

Answer 1

In the diagram, AB and CD represent the two chords of a circle with centre O. M and N are the midpoints of AB and CD, respectively, while MN is the line that connects the midpoints of two chords and passes through the centre O.

Since the straight line formed from the middle of a circle to bisect a chord is perpendicular to the chord,

∴ OM ⊥ AB and ON ⊥ CD.

So,

∠OMA = ∠OMB = 90° and ∠ONC = ∠OND = 90°

Since, ∠OMA = ∠OND = 90° (Alternate angles) and,

∠OMB = ∠ONC = 90° (Alternate angles)

Hence, proved that AB || CD.