[tex]\it \dfrac{BC}{AB}=\dfrac{\sqrt6}{3} \Rightarrow \dfrac{BC^2}{AB^2}=\dfrac{6}{9} \Rightarrow \dfrac{BC^2+AB^2}{AB^2}=\dfrac{6+9}{6} \stackrel{T.P.}{\Longrightarrow}\ \dfrac{AC^2}{AB^2}=\dfrac{15}{6} \Rightarrow\\ \\ \\ \Rightarrow \dfrac{(\sqrt{30})^2}{AB^2}=\dfrac{^{2)}15}{\ 6} \Rightarrow \dfrac{30}{AB^2}= \dfrac{30}{12} \Rightarrow AB^2=12 \Rightarrow AB=\sqrt{12}=2\sqrt3m[/tex]
[tex]\it \dfrac{BC}{AB}=\dfrac{\sqrt6}{3} \Rightarrow \dfrac{BC}{2\sqrt3}=\dfrac{\sqrt6}{3} \Rightarrow BC=\dfrac{2\sqrt3\cdot\sqrt6}{3}=\dfrac{2\sqrt3\cdot\sqrt6}{\sqrt3\cdot\sqrt3}=\dfrac{2\sqrt6}{\sqrt3}=2\sqrt2\\ \\ \\ \mathcal{A}=\dfrac{c_1\cdot c_2}{2}=\dfrac{AB\cdot BC}{2}=\dfrac{2\sqrt3\cdot2\sqrt2}{2}=2\sqrt6m^2[/tex]
