jmc

Let the side length of one of the squares be $x$. Looking at the right triangle with hypotenuse $AB$, we have the equation $x^2+(2x{)}^2=(2\sqrt{5}{)}^2$ from the Pythagorean Theorem. Simplifying this equation gives $x^2=4$. Looking at the right triangle with hypotenuse $AC$, we have the equation $x^2+(3x{)}^2=A{C}^2\Rightarrow A{C}^2=10x^2=40$. Thus, $AC=\sqrt{40}=\boxed{2\sqrt{10}}$ centimeters.