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

Prove that the line joining the mid-point of a chord to the centre of the circle passes through the mid-point of the corresponding minor arc.

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Given : ` M` is the midpoint of a chord `AB` of a circle `C(O,r)`, and `OM` is produced to meet the circle at `Q`.
To Prove : `hat(AQ) ~= hat(BQ)`
Construction : Join `OA` and `OB`
Proof : In `Delta OMA` and `Delta OMB`, we have
`OA = OB ` [ each equal to `r` ]
`OM` `=` `OM` [ common]
`AM` `=` `BM` [ `:'` `M` is the midpoint of `AB`]
`:. Delta OMA ~= Delta OMB` [ by `SSS`- congruence]
`implies /_ AOM =/_BOM` [ c.p.c.t.]
`implies m( hat(AQ))=m (hat(BQ))implies hat(AQ) ~= hat(BQ)`

Note : `MO` when produced to `P` also passes through the midpoint of the corresponding major arc.
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