Q. 64.4( 279 Votes )
Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
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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