Question

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

Answer 1

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.