The ratio of the areas of two triangles is 1:2 and the ratio of their bases is 3:4. What will be the ratio of their height?

Options

A. `1:3`
B. `4:3`
C. `2:1`
D. `2:3`

Answer

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

### Step-by-Step Solution:

1. **Understand the Given Ratios**:
- The ratio of the areas of the two triangles is given as \( A_1 : A_2 = 1 : 2 \).
- The ratio of the bases of the two triangles is given as \( B_1 : B_2 = 3 : 4 \).

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

3. **Set Up the Ratio of Areas**:
- From the area formula, we can express the ratio of the areas as:
\[ \frac{A_1}{A_2} = \frac{\frac{1}{2} B_1 H_1}{\frac{1}{2} B_2 H_2} = \frac{B_1 H_1}{B_2 H_2} \]
- Since we know \( \frac{A_1}{A_2} = \frac{1}{2} \), we can write:
\[ \frac{B_1 H_1}{B_2 H_2} = \frac{1}{2} \]

4. **Substitute the Base Ratios**:
- Substitute the known base ratios \( B_1 = 3k \) and \( B_2 = 4k \) (where \( k \) is a common factor):
\[ \frac{3k \cdot H_1}{4k \cdot H_2} = \frac{1}{2} \]
- The \( k \) cancels out:
\[ \frac{3 H_1}{4 H_2} = \frac{1}{2} \]

5. **Cross Multiply to Solve for Height Ratio**:
- Cross multiplying gives:
\[ 3 H_1 \cdot 2 = 4 H_2 \]
- Simplifying this:
\[ 6 H_1 = 4 H_2 \]
- Dividing both sides by 2:
\[ 3 H_1 = 2 H_2 \]

6. **Find the Ratio of Heights**:
- Rearranging gives:
\[ \frac{H_1}{H_2} = \frac{2}{3} \]

### Final Answer:
The ratio of the heights of the two triangles is \( H_1 : H_2 = 2 : 3 \).