The sum of the length and breadth of a rectangle is 6 cm. A square is constructed such that one of its sides is equal to a diagonal of the rectangle. If the ratio of areas of the square and rectangle is 5 : 2, the area of the square in ${\mathrm{cm}}^2$ is

Let length of rectangle = x then, 

Breath of rectangle = (6 - x)

$\because$Side of square= diagonal of rectangle. 

Let area of square = 5x 

Now,

$\begin{array}{l}{x}^2+(6-x{)}^2=5x\\ 2x^2-17x+36=0\\ 2x^2-9x-8x+36=0\\ x(2x-9)-4(2x-9)\end{array}$

$\begin{array}{l}(x-4)(2x-9)\\ x-4=0\\ \therefore x=4\\ 2x-9=0\\ \therefore x=9/2\end{array}$

Take x = 4; then,

$\therefore$ Area of square $=5x=5\times 4=20 {\mathrm{cm}}^2$.