ABCD is a square, F is mid-point of AB and BE is one third of BC. If area of △FBE is 108 cm², find the length of AC.

Let x cm be the side of each square.

Since, F is midpoint of AB so, FB = $\dfrac{x}{2}$ cm

BE is one third of BC, BE = $\dfrac{x}{3}$.

Given, area of △FBE is 108 cm²

$\therefore \dfrac{1}{2} \times FB \times BE = 108 \\[1em] \Rightarrow \dfrac{1}{2} \times \dfrac{x}{2} \times \dfrac{x}{3} = 108 \\[1em] \Rightarrow \dfrac{x^2}{12} = 108 \\[1em] \Rightarrow x^2 = 1296 \\[1em] \Rightarrow x = \sqrt{1296} = 36.$

Length of diagonal of square = $\sqrt{2}$ Side = $36\sqrt{2}$ cm.

Hence, length of diagonal of square = $36\sqrt{2}$ cm.