If the medians of two equilateral triangles are in the ratio 3:2, then what is ratio of the sides?

Options

A. `1:1`
B. `2:3`
C. `3:2`
D. `sqrt(3):sqrt(2)`

Answer

To solve the problem of finding the ratio of the sides of two equilateral triangles given that their medians are in the ratio of 3:2, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Medians**:
Let the side length of the first equilateral triangle be \( A_1 \) and its median be \( L_1 \). Similarly, let the side length of the second equilateral triangle be \( A_2 \) and its median be \( L_2 \).

2. **Use the Given Ratio**:
We know from the problem that the ratio of the medians is given as:
\[ \frac{L_1}{L_2} = \frac{3}{2} \]

3. **Formula for the Median of an Equilateral Triangle**:
The formula for the median \( L \) of an equilateral triangle with side length \( A \) is:
\[ L = \frac{\sqrt{3}}{2} A \]
Therefore, we can express the medians in terms of the sides:
\[ L_1 = \frac{\sqrt{3}}{2} A_1 \quad \text{and} \quad L_2 = \frac{\sqrt{3}}{2} A_2 \]

4. **Set Up the Ratio of Medians**:
Substitute the expressions for \( L_1 \) and \( L_2 \) into the ratio:
\[ \frac{L_1}{L_2} = \frac{\frac{\sqrt{3}}{2} A_1}{\frac{\sqrt{3}}{2} A_2} \]

5. **Simplify the Ratio**:
The \( \frac{\sqrt{3}}{2} \) terms cancel out:
\[ \frac{A_1}{A_2} = \frac{3}{2} \]

6. **Conclusion**:
Therefore, the ratio of the sides \( A_1 \) to \( A_2 \) is:
\[ A_1 : A_2 = 3 : 2 \]

### Final Answer:
The ratio of the sides of the two equilateral triangles is \( 3 : 2 \).