Question

ABCD is a rectangle and P,Q,R,S are mid-point of the sides AB,BC,CD and DA respectively . show that the quadrilateral PQRS is a rhombus.

Answer 1

when pq r and s are the mid point that means pq=rq=rs=spalso pq is parallel so it is a rhombus

Answer 2

Consider triangles PBQ,  QCR,  RDS,  and  SAP :
     AP = PB = CR = RD        They are all parallel to each other
     BC = CD = DS = SA            they are parallel to each other.

   angle PBQ = angle QCR = angle RDS = angle SAP = 90 deg.

  So all the four triangles are congruent.  So PQ = QR = RS = SP

   we know that QS   and  PR  are perpendicular to each other, as they are parallel to AB and BC respectively.

   hence, PQRS is a Rhombus.