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Prove that the line joining the mid-poin...

Prove that the line joining the mid-points of two parallel chords of a circle passes through the centre.

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Let AB and CD be two parallel chords having P and Q as their mid-points respectively. Let O be the centre of the circle. Join OP and OQ.
Now, P is the mid-points of AB.
`thereforeOPbot AB`
`rArr angle BPO = 90^@`
Similarly, Q is the mid-point of CD.
`rArrOQbot CD`
`rArrangle DQO=90^@`.
`therefore angleBPO+angle DQO=90^@+90^@=180^@`
Also, `AB||CD` (given)
`therefore POQ` must be a transversal i.e. a striaght line.
`rArrPQ` is a striaght line passing through O i.e. the centre of circle.
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