\(a,x:y:z=2:3:4\Rightarrow\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{4}\)
Đặt \(\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{4}=k\Rightarrow x=2k;y=3k;z=4k\)
\(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{13}{12}\Rightarrow\dfrac{1}{2k}+\dfrac{1}{3k}+\dfrac{1}{4k}=\dfrac{13}{12}\\ \Rightarrow\dfrac{6k^2+4k^2+3k^2}{12k}=\dfrac{13}{12}\\ \Rightarrow13k^2=13\\ \Rightarrow k^2=1\Rightarrow\left[{}\begin{matrix}k=1\\k=-1\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=2;y=3;z=4\\x=-2;y=-3;z=-4\end{matrix}\right.\)
\(b,\dfrac{AB}{3}=\dfrac{AC}{4}\Rightarrow\dfrac{AB^2}{9}=\dfrac{AC^2}{16}=\dfrac{AB^2+AC^2}{9+16}=\dfrac{BC^2}{25}=\dfrac{100}{25}=4\\ \Rightarrow\left\{{}\begin{matrix}AB^2=36\\AC^2=64\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}AB=6\left(cm\right)\\AC=8\left(cm\right)\end{matrix}\right.\left(AB,AC>0\right)\)
