\(y_{BC}=m.x+n\Rightarrow\left\{{}\begin{matrix}-5m+n=2\\-m+n=9\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}m=\dfrac{7}{4}\\n=\dfrac{43}{4}\end{matrix}\right.\)\(\Rightarrow y_{BC}=\dfrac{7}{4}x+\dfrac{43}{4}\)
\(M\in BC\Rightarrow M\left(x;\dfrac{7}{4}x+\dfrac{43}{4}\right)\)
\(\Rightarrow AM=\sqrt{\left(x-1\right)^2+\left(\dfrac{7}{4}x+\dfrac{23}{4}\right)^2}=\sqrt{x^2-2x+1+\dfrac{49}{16}x^2+\dfrac{161}{8}x+\dfrac{529}{16}}=\sqrt{\dfrac{65}{16}x^2+\dfrac{145}{8}x+\dfrac{545}{16}}\ge\sqrt{\dfrac{180}{13}}\)
\(AM_{min}\Leftrightarrow x=-\dfrac{29}{13}\Rightarrow M\left(-\dfrac{29}{13};\dfrac{89}{13}\right)\)
