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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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Given : `L` and `M` are midpoints of two parallel chords `AB` and `CD` respectively of a circle `C(O,r)` .
To prove : `LOM` is a straight line.
Construction : Join `OL, OM` and draw `OE` `||` `AB` `||` `CD` .
Proof : We know that the line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.
`:. OL _|_ AB` and `OM _|_ OE`
Now, `OL _|_ AB` and `AB || OE implies OL _|_ OE`
`implies /_ EOL =90^(@)`
`OM _|_ CD` and `CD || OE implies OM _|_ OE `
`implies /_ EOM =90^(@)`
`:. /_ EOL + /_ EOM =180^(@)`
Hence, `LOM` is a straight line.
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