# Q: 6     Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Given: ABCD is a  quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA. AC, BD are diagonals.

To prove: the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Proof: In ACD,

S is the midpoint of DA.                (Given)

R  is midpoint of DC.               (Given)

By midpoint theorem,

and   ...................................1

In ABC,

P is the midpoint of AB.                (Given)

Q  is the midpoint of BC.               (Given)

By midpoint theorem,

and   .................................2

From 1 and 2, we get

and

Thus,      and

So, the quadrilateral PQRS is a parallelogram and diagonals of a parallelogram bisect each other.

Thus, SQ and PR bisect each other.

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