The ratio of bases of two triangles is $x : y$ and that of their areas is $a : b$. Then the ratio of their corresponding altitudes will be

### Concept & Ratio of Areas of Triangles The area of a triangle is directly proportional to the product of its base and its altitude ($Area = \frac{1}{2} \times \text{base} \times \text{altitude}$). ### Step-by-Step Solution * Let the bases of the two triangles be $k \cdot x$ and $k \cdot y$. * Let their corresponding altitudes be $h_1$ and $h_2$. * The ratio of their areas is given as $\frac{a}{b}$. * Using the area formula: $$\frac{\text{Area}_1}{\text{Area}_2} = \frac{\frac{1}{2} \cdot (k \cdot x) \cdot h_1}{\frac{1}{2} \cdot (k \cdot y) \cdot h_2}$$ * Substitute the given area ratio: $$\frac{a}{b} = \frac{x \cdot h_1}{y \cdot h_2}$$ * Rearranging to solve for the ratio of the altitudes ($\frac{h_1}{h_2}$): $$\frac{h_1}{h_2} = \frac{a \cdot y}{b \cdot x}$$ * Therefore, the ratio is $ay : bx$. ### Exam Strategy & Shortcut Altitude is proportional to $\frac{\text{Area}}{\text{Base}}$. Therefore, the ratio of altitudes is simply $\frac{\text{Ratio of Areas}}{\text{Ratio of Bases}}$. Divide the area ratio terms by the base ratio terms: $\frac{a}{x} : \frac{b}{y}$, which simplifies to $ay : bx$ by cross-multiplying. ### Common Pitfall Accidentally multiplying the ratios directly to get $ax : by$ instead of dividing them. Always set up the standard area fraction first to prevent this. ### Final Answer Therefore, the correct answer is **$ay : bx$**.