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Question
Show that the line segments joining the mid-points of the opposite sides of a quadrilateral
bisect each other.
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Solution
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 ABD, S and P are the midpoints of AD and AB respectively.
So, by using midpoint theorem we can say that
SP || BD and SP = `1/2` BD ......(1)
Similarly in ΔBCD
QR || BD and QR = `1/2` BD .....(2)
From equation (1) and (2) we have
SP || QR and SP = QR
As in quadrilateral SPQR one pair 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.
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