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Question

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rhombus if

A
PQRS is a rhombus
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B
PQRS is a parallelogram
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C
diagonals of PQRS are perpendicular
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D
diagonals of PQRS are equal
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Solution

The correct option is D PQRS is a rhombus
Let us join AC and BD
In triangle ABC
P and Q are the mid point of AB ,AC respt
So PQ II AC and PQ=12AC..........(1)
Similarly triangle ADC
SR II AC and SR=12AC..........(2)
Clearly PQ II SR and PQ = SR
Since in quadrilateral PQRS one pair of side is equal and parallel
to each parallelogram
So PQ II QR and PS=PR ........................(3)
In triangle BCD Q and R are mid point of side BC and CD respt
So bQR II BD and QR=Qr=12BD..........(4)
However the diagonal of rectangle is equal
By using eq (1) ,(2),(3) and (4)
PQRS is a rhombus

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