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Question

# State true or falseP, Q, R, and S are the mid-point of the sides AB, BC, CD and DA of a quadrilateral ABCD such that AC ⊥ BD. Then PQRS is a rectangle

A
True
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B
False
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Solution

## The correct option is A TrueIn quadrilateral ABCD, P,Q,R and S are mid-points of sides AB,BC,CD and DA respectively.Since, AC⊥BD⇒ ∠COD=∠AOD=∠AOB=∠COB=90o.In △ADC,S and R the mid-points of AD and DC respectively.Then, by mid-point theorem⇒ SR∥AC and SR=12AC ---- ( 1 )In △ABC,P and Q are the mid-points of AB and BC respectively, then by mid-point theorem⇒ PQ∥AC and PQ=12AC --- ( 2 )From ( 1 ) and ( 2 ),⇒ PQ∥SR and PQ=SR=12AC ---- ( 3 )Similarly, SP∥RQ and SP=RQ=12BD ----- ( 4 )Now, in quadrilateral EOFR, OE∥FR, OF∥ER⇒ ∠EOF=∠ERF=90o [ Since ∠COD=90o⇒∠EOF=90o ] --- ( 5 )From ( 3 ), ( 4 ) and ( 5 ) we can prove that,∴ PQRS is rectangle.∴ The given statement is correct.

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