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Question -

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.



Answer -

Let P is the mid point of chord AB of circle C(O, r) then according to question, line OQ passes through the point P.
Then prove that OQ bisect the arc AB.
Join OA and OB.
In тЦ│AOP and тЦ│BOPтЦ│AOP and тЦ│BOP
┬а ┬а ┬а┬а┬а ┬а ┬а ┬а ┬а ┬а ┬а ┬а ┬а ┬а (Radii of the same circle)
┬а ┬а┬а┬а ┬а ┬а ┬а ┬а ┬а ┬а ┬а ┬а (P is the mid point of chord AB)
┬а ┬а ┬а┬а ┬а ┬а ┬а ┬а (Common)
Therefore,┬а┬а
┬а ┬а ┬а┬а┬а ┬а ┬а ┬а ┬а ┬а ┬а ┬а(by cpct)
Thus
Arc AQ = arc BQ
Therefore,┬а┬а
Hence Proved.

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