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

Options

A. `(a)/(x) : (b)/(y)`
B. `a x : by`
C. `a y : b x`
D. `(x)/(a) : (b)/(y)`

Answer

To solve the problem, we need to find the ratio of the corresponding altitudes of two triangles given the ratio of their bases and the ratio of their areas.

### Step-by-step solution:

1. **Define the Variables**:
Let the base of triangle 1 be \( b_1 = x \) and the base of triangle 2 be \( b_2 = y \).
Let the altitude of triangle 1 be \( h_1 \) and the altitude of triangle 2 be \( h_2 \).

2. **Area of the Triangles**:
The area of triangle 1 can be expressed as:
\[ A_1 = \frac{1}{2} \times b_1 \times h_1 = \frac{1}{2} \times x \times h_1 \]
The area of triangle 2 can be expressed as:
\[ A_2 = \frac{1}{2} \times b_2 \times h_2 = \frac{1}{2} \times y \times h_2 \]

3. **Given Ratios**:
We are given that the ratio of the areas of the two triangles is \( A_1 : A_2 = a : b \). Therefore, we can write:
\[ \frac{A_1}{A_2} = \frac{a}{b} \]

4. **Substituting the Areas**:
Substituting the expressions for the areas into the ratio gives us:
\[ \frac{\frac{1}{2} \times x \times h_1}{\frac{1}{2} \times y \times h_2} = \frac{a}{b} \]
The \( \frac{1}{2} \) cancels out:
\[ \frac{x \times h_1}{y \times h_2} = \frac{a}{b} \]

5. **Cross Multiplying**:
Cross-multiplying gives us:
\[ x \times h_1 \times b = y \times h_2 \times a \]

6. **Rearranging for Altitudes**:
Rearranging the equation to find the ratio of the altitudes \( h_1 \) and \( h_2 \):
\[ \frac{h_1}{h_2} = \frac{y \times a}{x \times b} \]

7. **Final Ratio**:
Thus, the ratio of the corresponding altitudes is:
\[ h_1 : h_2 = ya : xb \]

### Conclusion:
The ratio of the corresponding altitudes of the two triangles is \( \frac{ya}{xb} \).