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Question 4
P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD such that ACBD. Prove that PQRS is a rectangle.


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Solution

Given In quadrilateral ABCD, P, Q, R and S are the mid-points of the sides AB, BC, CD and DA, respectively.

Also, ACBD

To prove PQRS is a rectangle.

Proof since, ACBD

COD=AOD=AOB=COB=90

In ΔADC, S and R are the mid-points of AD and DC respectively, then by mid-point theorem

SR||AC and SR=12AC ………………..(i)

In ABC, P and Q are the mid-points of AB and BC respectively, then by mid-point theorem,

PQAC and PQ12=AC ……(ii)

From Eqs.(i) and (ii), PQSR and PQ=SR=12AC ….(iii)

Similarly SP ∥ RQ and SP=RQ=12BD …..(iv)

Now, in quadrilateral EOFR, OE ∥FR, OF ∥ ER

EOF=ERF=90[COD=90EOF=90] ….(v)

So, PQRS is a rectangle. Hence proved.


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