A line joining the midpoints of two chords pa | vedclass

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(A) Let $AB$ and $CD$ be two chords of a circle with centre $O$. Let $M$ and $N$ be the midpoints of chords $AB$ and $CD$ respectively.Given that the

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The chords are parallel.

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(A) Let $AB$ and $CD$ be two chords of a circle with centre $O$. Let $M$ and $N$ be the midpoints of chords $AB$ and $CD$ respectively.Given that the line $MN$ passes through the centre $O$.Since $M$ is the midpoint of $AB$,the line segment joining the centre to the midpoint of a chord is perpendicular to the chord. Therefore,$OM \perp AB$,which implies $\angle OMA = 90^{\circ}$.Similarly,since $N$ is the midpoint of $CD$,$ON \perp CD$,which implies $\angle ONC = 90^{\circ}$.Since the points $M, O, N$ are collinear (as the line $MN$ passes through the centre $O$),the line $MN$ acts as a transversal to the chords $AB$ and $CD$.The angles $\angle OMA$ and $\angle ONC$ are corresponding angles (or interior angles on the same side depending on the orientation). Specifically,$\angle OMA + \angle ONC = 90^{\circ} + 90^{\circ} = 180^{\circ}$ if they are interior angles on the same side.Since the sum of the interior angles on the same side of the transversal $MN$ is $180^{\circ}$,the lines $AB$ and $CD$ must be parallel.

$A$ line joining the midpoints of two chords passes through the centre of a circle. Prove that the chords are parallel.

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