Problem - 2984
In $\triangle{ABC}$, if $(b+c):(c+a):(a+b)=5:6:7$, compute $\sin{A}:\sin{B}:\sin{C}$.
Let $b+c=5k, c+a=6k,$ and $a+b=7k$ where $k$ is a constant. Solving this system of three equations gives us $$a= 4k, b=3k, c=2k\quad\text{or}\quad a:b:c=4:3:2$$
Thus, by the law of sine we have $$\sin{A}:\sin{B}:\sin{C}=4:3:2$$