Gọi AB = 2x, AC = 3x

Theo hệ thức lượng trong tam giác vuông ta có:

$\frac{1}{A{{B}^{2}}}+\frac{1}{A{{C}^{2}}}=\frac{1}{A{{H}^{2}}}\Leftrightarrow \frac{1}{4{{x}^{2}}}+\frac{1}{9{{x}^{2}}}=\frac{1}{36}$

$\Leftrightarrow \frac{9}{36{{x}^{2}}}+\frac{4}{36{{x}^{2}}}=\frac{1}{36}\Leftrightarrow \frac{13}{36{{x}^{2}}}=\frac{1}{36}$

$\Leftrightarrow \frac{13}{36{{x}^{2}}}=\frac{{{x}^{2}}}{36{{x}^{2}}}\Leftrightarrow {{x}^{2}}=13\Rightarrow x=\sqrt{13}$

$AC=3x=3\sqrt{13}cm$