Question

# P,Q,R,R, and S are respectively the mid-points of sides AB, BC, CD, And DA of a quadrilateral ABCD in which AC =BD, then PQRS is a rhombus.

A
True
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B
False
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Solution

## The correct option is A TrueP,Q,R and S are the mid-point of the sides AB,BC,CD and DA of a quadrilateral ABCD.⇒ AC=BDIn △ABC,P and Q are the mid-points of the sides AB and BC respectively.∴ PQ∥AC ----- ( 1 )And PQ=12×AC ------ ( 2 )Similarly, SR∥AC and SR=12×AC ----- ( 3 )From ( 1 ), ( 2 ) and ( 3 ) we get,⇒ PQ∥SR and PQ=SR=12×AC ----- ( 4 )Similarly we an show that,⇒ SP∥RQ and SP=RQ=12×BD ----- ( 5 ) Since, AC=BD∴ PQ=SR=SP=RQ [ From ( 4 ) and ( 5 ) ]All sides of the quadrilateral are equal.∴ PQRS is a rhombus.

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