Đáp án:
$\begin{array}{l}
a)A{B^2} + A{C^2} = {5^2} + {12^2} = 169\\
B{C^2} = {13^2} = 169\\
\Rightarrow A{B^2} + A{C^2} = B{C^2}\\
\Rightarrow \Delta ABC \bot A\left( {theo\,Pytago\,dao} \right)\\
b)Theo\,t/c:\\
\dfrac{{BD}}{{AB}} = \dfrac{{CD}}{{AC}}\\
\Rightarrow \dfrac{{BD}}{5} = \dfrac{{CD}}{{12}} = \dfrac{{BD + CD}}{{5 + 12}} = \dfrac{{BC}}{{17}} = \dfrac{{13}}{{17}}\\
\Rightarrow \left\{ \begin{array}{l}
BD = \dfrac{{65}}{{17}}\left( {cm} \right)\\
CD = \dfrac{{156}}{{17}}\left( {cm} \right)
\end{array} \right.
\end{array}$