\[\begin{array}{l}
Ta\,\,\,co:\\
\frac{{AM}}{{AB}} = \frac{{10}}{{25}} = \frac{2}{5}\\
\frac{{AN}}{{AC}} = \frac{{12}}{{30}} = \frac{2}{5}\\
\Rightarrow \frac{{AM}}{{AB}} = \frac{{AN}}{{AC}} = \frac{2}{5}\\
\Rightarrow MN//BC\,\,\left( {dinh\,\,ly\,\,\,Ta - let\,\,dao} \right)\\
\Rightarrow \frac{{MN}}{{BC}} = \frac{2}{5}\\
\Rightarrow BC = \frac{{5.MN}}{2} = \frac{{5.14}}{2} = 35\,\,cm.
\end{array}\]