Đáp án:
$\begin{array}{l}
a)\frac{{AM}}{{AB}} = \frac{2}{7}\\
\frac{{AN}}{{AC}} = \frac{{AN}}{{AN + NC}} = \frac{3}{{3 + 7,5}} = \frac{2}{7}\\
\Rightarrow \frac{{AM}}{{AB}} = \frac{{AN}}{{AC}} = \frac{2}{7}\\
\Rightarrow MN//BC\left( {Theo\,Talet} \right)
\end{array}$
$\begin{array}{l}
b)\frac{{BD}}{{AB}} = \frac{{CD}}{{AC}}\\
\Rightarrow \frac{{BD}}{7} = \frac{{CD}}{{10,5}} = \frac{{BD + CD}}{{17,5}} = \frac{{BC}}{{17,5}} = \frac{{14}}{{17,5}} = \frac{4}{5}\\
\Rightarrow \left\{ \begin{array}{l}
BD = \frac{{28}}{5} = 5,6\left( {cm} \right)\\
CD = 8,4\left( {cm} \right)
\end{array} \right.
\end{array}$