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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.
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Solution
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.
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