Question

P q r s are respectively the midpoints of sides ab,bc,cd,da respectively of a quadrilateral abcd in which ab=cd.prove that pqrs is a rhombus?

Answer 1

P,Q,R and S are respectively the mid point of the sides AB,BC,CD and DA of a quadrilateral ABCD in which AD=BC.

In the triangle ABC;
P and Q are the mid-points of the sides AB and BC respectively.
therefore PQ || AC........(1)
and PQ = 1/2*AC.......(2)
similarly SR || AC and SR =1/2*AC......(3)
therefore PQ || SR and PQ = SR=1/2*AC.............4(a)
similarly we can show that SP || RQ and SP = RQ=1/2*BD........4(b)
since AC = BD
therefore PQ = SR = SP = RQ
all the sides of the quadrilateral are equal.
Therefore PQRS is a rhombus.