Given that the ratio of altitudes of two triangles is 4 :5, ratio of their areas is 3 : 2. The ratio of their corresponding bases is

Options

A. `8 : 15`
B. `15 : 8`
C. `5 : 8`
D. `8 : 5`

Answer

To find the ratio of the corresponding bases of two triangles given the ratio of their altitudes and the ratio of their areas, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Ratios**:
- Let the ratio of the altitudes of the two triangles be \( H_1 : H_2 = 4 : 5 \).
- Let the ratio of the areas of the two triangles be \( A_1 : A_2 = 3 : 2 \).

2. **Use the Formula for Area of a Triangle**:
- The area of a triangle can be expressed as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
- For the two triangles, we can write:
\[ A_1 = \frac{1}{2} \times B_1 \times H_1 \quad \text{and} \quad A_2 = \frac{1}{2} \times B_2 \times H_2 \]

3. **Set Up the Ratio of Areas**:
- From the area formulas, we can express the ratio of the areas as:
\[ \frac{A_1}{A_2} = \frac{B_1 \times H_1}{B_2 \times H_2} \]

4. **Substitute the Known Ratios**:
- We know that \( \frac{A_1}{A_2} = \frac{3}{2} \) and \( \frac{H_1}{H_2} = \frac{4}{5} \).
- Substitute these values into the area ratio:
\[ \frac{3}{2} = \frac{B_1 \times H_1}{B_2 \times H_2} \]
- This can be rearranged to:
\[ \frac{3}{2} = \frac{B_1}{B_2} \times \frac{H_1}{H_2} \]

5. **Substitute the Height Ratio**:
- Substitute \( \frac{H_1}{H_2} = \frac{4}{5} \) into the equation:
\[ \frac{3}{2} = \frac{B_1}{B_2} \times \frac{4}{5} \]

6. **Solve for the Ratio of Bases**:
- Rearranging gives:
\[ \frac{B_1}{B_2} = \frac{3}{2} \times \frac{5}{4} \]
- Calculate:
\[ \frac{B_1}{B_2} = \frac{3 \times 5}{2 \times 4} = \frac{15}{8} \]

7. **Conclusion**:
- The ratio of the corresponding bases \( B_1 : B_2 \) is \( 15 : 8 \).

### Final Answer:
The ratio of the corresponding bases is \( 15 : 8 \).