10.084. The perimeter of a rhombus is 2 m, the lengths of its diagonals are in the ratio $3: 4$. Find the area of the rhombus.
Solution.

The desired area will be found using the formula $S=\frac{1}{2} A C \cdot B D=2 A O \cdot O B$ (Fig.

10.81). In $\triangle A O B, A B^{2}=A O^{2}+B O^{2}$, where $O B=\frac{3}{4} A O, A B=\frac{2}{4}=\frac{1}{2}$ (m). Then we get the equation $\frac{1}{4}=A O^{2}+\frac{9}{16} A O^{2}$, from which $A O^{2}=\frac{16}{100}$, i.e., $A O=0.4$ (m). Thus, $S=2 \cdot 0.4 \cdot 0.3=0.24\left(\mathrm{~m}^{2}\right)$.

Answer: $0.24 \mathrm{~m}^{2}$.