ABCD is a rectangle and PQRS the figure formed by joining in order the middle points of the sides.prove that 1)PQRS is a rhombus 2) area of PQRS =1/2 AREA of ABCD

AC and BD are the diagonals of rectangle.

 

Now, PS = 1/2 (AC) = QR

 

And, PQ = 1/2 (BD) = RS

 

But, Ac = BD (diagonals of a rectangle are equal)

 

Thus, PQ = QR = RS = PS

 

Thus, PQRS is a rhombus.

 

Now, area of rhombus PQRS = 1/2 (product of diagonals)

 

Thus, area PQRS = 1/2 x (PR) x (QS)

 

But PR = AD = BC  and QS = AB = CD

 

And, area of rectangle = base x height = AD x CD

 

Thus, area of PQRS = 1/2 x (AD) x (CD) = 1/2 (area of rectangle ABCD)