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Question

# In the quadrilateral given below, AD=BC.P,Q,R and S are mid-points of AB,BD,CD and AC respectively. Prove that PQRS is a rhombus.

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Solution

## AD=BC [ Given ]The mid-point theorem states that the segment joining two sides of triangle at the mid-points of those sides is parallel to the third side and is half the length of the third side.In △BAD,P and S are the mid-points of sides AB and BD.So, By mid-point theorem,⇒ PS∥AD and PS=12AD ------- ( 1 )In △CAD,R and Q are the mid-point of CD and AC.So, by mid-point theorem,⇒ OR∥AD and QR=12AD ----- ( 2 )Compare ( 1 ) and ( 2 ), we get⇒ PS∥QR and PS=QRSince, one pair of opposite sides is equal as well as parallel then,⇒ PQRS is a parallelogram ---- ( 3 )Now, In △ABC, by mid-point theorem⇒ PQ∥BC and PQ=12BC ----- ( 4 )And AD=BC ----- ( 5 )Compare equations ( 1 ), ( 4 ) and (5) we get,⇒ PS=PQ ----- ( 6 )Since, PQRS is a parallelogram with PS=PQ then PQRS is a rhombus.

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