ABC26GN4493 · Area

Let the sides of the triangle be $2 x, 3 x, 4 x$ and their corresponding altitudes be $h_{1}, h_{2}, h_{3}$ respectively. Then, $$\begin{aligned} & \frac{1}{2} \times 2 x \times h_{1}=\frac{1}{2} \times 3 x \times h_{2}=\frac{1}{2} \times 4 x \times h_{3} \\ & \Rightarrow x h_{1}=\frac{3}{2} x h_{2}=2 x h_{3}=k(\text { say }) \\ & \Rightarrow h_{1}=\frac{k}{x}, h_{2}=\frac{2 k}{2 x}, h_{3}=\frac{k}{2 x} . \\ & \therefore h_{1}: h_{2}: h_{3}=\frac{k}{x}: \frac{2 k}{3 x}: \frac{k}{2 x}=1: \frac{2}{3}: \frac{1}{2}=6: 4: 3 . \end{aligned}$$