10.118. Determine the side of the rhombus, knowing that its area is $S$, and the lengths of the diagonals are in the ratio $m: n$.
Solution.

Let's write $B D=m x ; A C=n x$ (Fig. 10.111). The area of the rhombus $S=\frac{1}{2} m n x^{2}$, from which $x=\sqrt{\frac{2 S}{m n}} \cdot$ Then $B D=m \sqrt{\frac{2 S}{m n}}=\sqrt{\frac{2 S m}{n}} ; A C=\sqrt{\frac{2 S n}{m}}$. The side of the rhombus $B C^{2}=\frac{1}{4} B D^{2}+\frac{1}{4} A C^{2}==\frac{2 S m}{4 n}+\frac{2 S n}{4 m}=\frac{S\left(m^{2}+n^{2}\right)}{2 m n}$. From this, $B C=\sqrt{\frac{S\left(m^{2}+n^{2}\right)}{2 m n}}$.

Answer: $\sqrt{\frac{S\left(m^{2}+n^{2}\right)}{2 m n}}$