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Show that the line segments joining th...

Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

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Let ABCD is a quadrilateral and P,Q,R and S are the mid-points of the sides AB,BC,Co and DA, respectively. i.e., AS = AP= BP, BQ=CQ and CR = DR. We have to show have to show that PR and SQ bisect each other i.e., SO= OQ and PO = OR.
Now, in `DeltaABD,S and R` are mid-points of AD and CD. We known that, the line seginent joining jthe mid-points of two sides of a triangles is parallel to the third side.

`therefore" "SR"||"AC and SR=1/2AC" "("by mid-point theorem")...(1)`
Similarly, in `DeltaABC,P and Q` are mid-points of AB and BC
`therefore" "PQ"||"ACand PQ=1/2AC" "("by mid-point theorem")...(2)`
From (1) and (2) we, get
`PQ"||"SRand PQ=SR=1/2AC`
`therefore` Quadrilateral PQRS is a parallelogram whose diagonals are SQ and PR. Also we know that diagonals of a parallelogram bisect each other. So, SQ and PR bisect each other.
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