Mathematics
The line joining mid-points of two chords of a circle passes through its center. Prove that the chords are parallel.
Answer
In the figure, AB and CD are the two chords of a circle with center O. M and N are mid-points of AB and CD, respectively and MN is the line joining the mid-points of two chords and passing through center O.
Since, the straight line drawn from the centre 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.
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