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 AC⊥BD. Prove that PQRS is a rectangle.
Given In quadrilateral ABCD, P, Q, R and S are the mid-points of the sides AB, BC, CD and DA, respectively.
Also, AC⊥BD
To prove PQRS is a rectangle.
Proof since, AC⊥BD
∴∠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,
PQ∥AC and PQ12=AC ……(ii)
From Eqs.(i) and (ii), PQ∥SR 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=90∘⇒∠EOF=90∘] ….(v)
So, PQRS is a rectangle. Hence proved.