Question

show that the line segments joining the mid points of opposite sides of quadrilateral bisect each other.

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Answer 1

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Answer 2

let ABCD is a quadrilateral in which P, Q, R and S are midpoints of sides AB , BC , CD and DA respectively join PQ , QR , RS, SP and BD .

In triangle S and P are the midpoints of AD and AB respectively.

So by using midpoint theorem we can say that -

SP || BD and SP = ½ BD ---------- (i)

Similarly in Triangle ABCD

QR || BD and QR = ½ BD ---------- (ii)

From Equation ( i )and ( ii ) we have

SP || QR = SP || QR

As in quadrilateral SPQR one point of opposite side are equal and parallel to each other.

So SPQR is parallelogram

Since diagonals of a parallelogram bisect each other.

Hence PR and QS bisect each other.

[ Note :- please refer to the picture also ]