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