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ABCD is quadrilateral in which . If P,Q,R,S be the mid-points of AB, AC, CD and BD respectively, show that PQRS is a rhombus.

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answer is A.

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Detailed Solution

In △BAD, P and S are mid-points of AB and BD
⇒ PS ∥ AD and PS=21AD --- ( 1 ) [ By BPT ]
Similarly in △CAD,
⇒ OR ∥ AD and QR=

AD --- ( 2 )
From ( 1 ) and ( 2 ), we get
⇒ PS ∥ QR and PS=QR=

AD ------ ( 3 )
In △BDC, we get
⇒ SR ∥ BC and SR=

BC ----- ( 4)
And in △ABC
PQ ∥ BC and PQ=

BC ------ ( 5 )
⇒ PQ ∥ SR, PQ=SR=

BC ---- ( 6 ) [ From ( 4 ) and ( 5 ) ]
∴ □PQRS is a parallelogram.
Now, AD=BC
∴

AD=

BC
∴ PS=QR=PQ=SR [ From ( 3 ) and ( 6 ) ]
∴ □PQRS is a rhombus.
Hence, as we know rhombus has equal four sides and parallel opposite sides.
We have that parallelogram PQRS is rhombus as all four sides or it are equal

So, option 1 is correct.
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