2. Given that the three altitudes of $\triangle A B C$ are $A D=3, B E=4, C F=5$, and the lengths of the three sides of this triangle are all integers, then the minimum value of the length of the shortest side is
A. 10
B. 12
C. 14
D. 16
2. B From $S_{\triangle A B C}=\frac{1}{2} B C \cdot A D=\frac{1}{2} C A \cdot B E=\frac{1}{2} A B \cdot C F$ we get $3 B C=4 C A=5 A B$. It is evident that $A B$ is the shortest side. From $B C=\frac{5}{3} A B, C A=\frac{5}{4} A B$ and the fact that the lengths of $B C, C A, A B$ are all integers, we know that $3 \mid A B$ and $4 \mid A B$. Since $(3,4)=12$, it follows that $12 \mid A B$. Therefore, the minimum possible length of the shortest side $A B$ is 12.