[tex]\Delta ABC; mas\ \textless \ C=90^0 \\\\ sin\ \textless \ B=\frac{2}{3} \\\\ sin\ \textless \ B=\frac{AC}{AB} \\\\\\ \Rightarrow \frac{AC}{AB}=\frac{2}{3} \\\\ \frac{AC}{2}=\frac{AB}{3}=k \\\\\\ AC \to 2k \\ AB\to 3k \\\\ BC^2=AB^2-AC^2 \\\\ BC^2=(3k)^2-(2k)^2 \\\\ BC^2=9k^2-4k^2 \\\\ BC^2=5k^2 \\\\ \underline{BC \to k\sqrt{5}} \\\\\\\\ \frac{AB}{BC}\longrightarrow \frac{3 \not k}{\not k\sqrt{5}}= \frac{3}{\sqrt{5}} \to \frac{3\sqrt{5}}{5} \\\\\\ \boxed{\frac{AB}{BC} \to \frac{3\sqrt{5}}{5}}[/tex]
