Q. 64.4( 279 Votes )

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

Answer :

Mid point theorem:The line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the third side.
Let ABCD is a quadrilateral in which P, Q, R, and S are the mid-points of sides AB, BC, CD, and DA respectively. Join PQ, QR, RS, SP, and BD


In ΔABD, S and P are the mid-points of AD and AB respectively. Therefore, by using mid-point theorem:


SP || BD and SP = BD (1)


Similarly in ΔBCD,


QR || BD and QR = BD (2)


From equations (1) and (2), we obtain


SP || QR and SP = QR


In quadrilateral SPQR, one pair of opposite sides is equal and parallel to each other


Therefore, SPQR is a parallelogram.


We know that diagonals of a parallelogram bisect each other


Hence, PR and QS bisect each other.

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