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

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

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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 \triangleACD,     

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 \triangleABC,

      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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