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Question
Prove that the line segment joining the midpoints of two parallel chords of a circle passes through its centre.
Solution
Given: AB and CO are two chords of a cirde with centre O.
AB II CD , M and N are midpoints of AB and CO respectively.
To prove : MN passes through centre O.
Construction : Join OM, ON, and through O, draw a straight line EF parallel to AB.
Proof : OM ⊥ AB
(line joining the midpoin t of a chord to the centre of a circle is perpendicular to it)
∠ AMO = 90°
∠ MOE = 90° [cointerior angle of ∠ AMO]
∠ NOE = 90° [corresponding angle of ∠ AMO]
∠ MOE + ∠ NOE = 180°
∠ MON is a straight line .
Hence, MN passes through centre O.
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