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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.

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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 that AB || CD

Proof
L is the mid-point of AB as OP is perpendicular bisector of AB.
OLAB
[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, OMCD
OMD=90....(ii)
From Eqs. (i) and (ii), ALO=OMD=90
But, these are alternate angles
So AB || CD, hence proved.


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