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.
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.
∴ 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 Eqs. (i) and (ii), ∠ALO=∠OMD=90∘
But, these are alternate angles
So AB || CD, hence proved.