[tex]\frac{c_1}{c_2}=\sqrt2 \\ m_i=3 \ cm =\frac{i}{2} \\ \Rightarrow i=3 \ cm \cdot 2=6 \ cm \\ c_1^2+c_2^2=i^2=36 \ cm^2\\ c_1=c_2\sqrt2 \\ (c_2\sqrt2)^2+c_2^2=2c_2^2+c_2^2=3c_2^2=36 \ cm^2 \\ \Rightarrow \ c_2^2=\frac{36 \ cm^2}{3}=12 \ cm^2 \ \Rightarrow c_2=2\sqrt3 \ cm \\ \Rightarrow c_1=2\sqrt3 \ cm \cdot \sqrt2=2\sqrt6 \ cm \\ \Rightarrow \ A_{\Delta}=\frac{c_1\cdot c_2}{2}=\frac{2\sqrt6 \ cm\cdot 2\sqrt3 \ cm}{2}=\boxed{6\sqrt{2} \ cm^2}[/tex]
