Special Right Triangles

\begin{equation*}
\begin{split}
\text{AB} : \text{AC} & = 1 : \sqrt{2}\\\\
\text{AB} : 10 \text{ cm}& = 1 : \sqrt{2}\\\\
\frac{\text{AB}}{10}& = \frac{1}{\sqrt{2}}\\\\
\text{AB}& = \frac{10}{\sqrt{2}}\times \color{red} \frac{\sqrt{2}}{\sqrt{2}}\\\\
\text{AB}& = \frac{10\sqrt{2}}{2}\\\\
\text{AB}& = 5\sqrt{2} \text{ cm}\\\\
\end{split}
\end{equation*}