If the ratio of the altitudes of two triangles be 3:4 and the ratio of their corresponding areas be 4 : 3, then the ratio of their corresonding lengths of bases is

Options

A. `1:1`
B. `16:9`
C. `1:2`
D. `2:1`

Answer

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

### Step-by-Step Solution:

1. **Understand the Given Ratios**:
- The ratio of the altitudes of the two triangles is given as \( H_1 : H_2 = 3 : 4 \).
- The ratio of their corresponding areas is given as \( A_1 : A_2 = 4 : 3 \).

2. **Use the Area Formula**:
- The area of a triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
- For the first triangle:
\[ A_1 = \frac{1}{2} \times B_1 \times H_1 \]
- For the second triangle:
\[ A_2 = \frac{1}{2} \times B_2 \times H_2 \]

3. **Set Up the Ratio of Areas**:
- From the area formulas, we can express the ratio of the areas as:
\[ \frac{A_1}{A_2} = \frac{\frac{1}{2} \times B_1 \times H_1}{\frac{1}{2} \times B_2 \times H_2} = \frac{B_1 \times H_1}{B_2 \times H_2} \]
- This simplifies to:
\[ \frac{B_1 \times H_1}{B_2 \times H_2} = \frac{4}{3} \]

4. **Substitute the Ratio of Heights**:
- We know \( \frac{H_1}{H_2} = \frac{3}{4} \). Substitute this into the area ratio:
\[ \frac{B_1 \times \frac{3}{4} H_2}{B_2 \times H_2} = \frac{4}{3} \]
- Cancel \( H_2 \) from both sides:
\[ \frac{B_1 \times 3}{B_2 \times 4} = \frac{4}{3} \]

5. **Cross Multiply**:
- Cross multiplying gives:
\[ 3B_1 \times 3 = 4B_2 \times 4 \]
- This simplifies to:
\[ 9B_1 = 16B_2 \]

6. **Find the Ratio of Bases**:
- Rearranging gives:
\[ \frac{B_1}{B_2} = \frac{16}{9} \]

### Final Answer:
The ratio of the corresponding lengths of bases is \( B_1 : B_2 = 16 : 9 \).

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