If the sides of a triangle be in the ratio 2 : 3 : 4, the ratio of the corresponding altitudes is

### Concept & Inverse Proportionality of Altitudes For any given triangle, the area is constant regardless of which side is chosen as the base. Since $Area = \frac{1}{2} \times \text{base} \times \text{altitude}$, the altitude is inversely proportional to its corresponding base length. ### Step-by-Step Solution * Let the area of the triangle be $A$. * Let the sides (bases) be $2x$, $3x$, and $4x$. * The corresponding altitudes are $h_1$, $h_2$, and $h_3$. * Since area is constant: $\frac{1}{2}(2x)h_1 = \frac{1}{2}(3x)h_2 = \frac{1}{2}(4x)h_3 = A$. * Therefore, $h_1 \propto \frac{1}{2}$, $h_2 \propto \frac{1}{3}$, and $h_3 \propto \frac{1}{4}$. * The ratio of the altitudes is $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$. * To simplify this into whole numbers, multiply by the Least Common Multiple (LCM) of the denominators (2, 3, and 4), which is 12. * $\frac{12}{2} : \frac{12}{3} : \frac{12}{4} = 6 : 4 : 3$. ### Exam Strategy & Shortcut When sides are given as a ratio $a : b : c$, the ratio of their corresponding altitudes is simply $\frac{1}{a} : \frac{1}{b} : \frac{1}{c}$. Find the LCM of $a, b, c$ and multiply it through to get the final integer ratio instantly. ### Common Pitfall Reversing the direct ratio to $4 : 3 : 2$. Inverse ratios of three numbers do not work by simply reversing the order; you must use reciprocals. ### Final Answer Therefore, the correct answer is **6 : 4 : 3**.