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

M mansi

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 $\triangle$ACD,

S is the midpoint of DA.                (Given)

R  is midpoint of DC.               (Given)

By midpoint theorem,

$\small SR\parallel AC$  and   $\small SR=\frac{1}{2}AC$...................................1

In $\triangle$ABC,

P is the midpoint of AB.                (Given)

Q  is the midpoint of BC.               (Given)

By midpoint theorem,

$\small PQ\parallel AC$  and   $\small PQ=\frac{1}{2}AC$.................................2

From 1 and 2, we get

$\small PQ\parallel SR$          and   $\small PQ=SR=\frac{1}{2}AC$

Thus, $\small PQ=SR$     and $\small PQ\parallel SR$

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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