Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
ABCD is a quadrilateral. P, Q, R and S are mid-points of sides AB, BC, CD and DA respectively.
We need to prove that OP=OR and OS=OQ. Join AC.
In , S is the mid-point of AD and R is the mid-point of DC.
Therefore, by mid-point theorem, we have and (1)
In , Q is the mid-point of BC and P is the mid-point of AB.
Therefore, by mid-point theorem, we have and (2)
From (1) and (2), we have and
is a parallelogram.
(A quadrilateral is a parallelogram if one pair of opposite sides is equal and parallel)
and (Diagonals of parallelogram bisect each other)