If the perimeter of a rectangular plot is 68 m and length of its diagonal is 26 m , find its area.

Taking length = l and breadth = b

Perimeter of rectangle = 2(l + b)

Length of diagonal of a rectangle = $\sqrt{l^2 + b^2}$

Given,

Perimeter = 68 m

$\Rightarrow 2(l + b) = 68 \\[0.5em] \Rightarrow l + b = 34 \\[0.5em] \Rightarrow l = 34 - b \qquad \text{ Equation (a)}$

Given,

Diagonal of a rectangle = 26 m

$\Rightarrow \sqrt{l^2 + b^2} = 26$

On squaring both sides,

$\Rightarrow l^2 + b^2 = 26^2$

Putting values of l from equation a,

$\Rightarrow (34 - b)^2 + b^2 = 676 \\[1em] \Rightarrow 1156 + b^2 - 68b + b^2 = 676 \\[1em] \Rightarrow 2b^2 - 68b + 1156 - 676 = 0 \\[1em] \Rightarrow 2b^2 - 68b + 480 = 0 \\[1em] \Rightarrow 2(b^2 - 34b + 240) = 0 \\[1em] \Rightarrow b^2 - 34b + 240 = 0 \\[1em] \Rightarrow b^2 - 24b - 10b + 240 = 0 \\[1em] \Rightarrow b(b - 24) - 10(b - 24) = 0 \\[1em] \Rightarrow (b - 24)(b - 10) = 0 \\[1em] \Rightarrow b - 24 = 0 \text{ or } b - 10 = 0 \\[1em] b = 24 \text{ or } b = 10$

∴ If b = 24 ,l = 34 - b = 34 - 24 = 10

If b = 10 , l = 34 - b = 34 - 10 = 24

Area of rectangle = Length $\times$ Breadth = 24 $\times$ 10 = 240 m²

Hence, the area of rectangle is 240 m².