# 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.

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by vibhu25

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by vibhu25

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

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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.

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.