The ratio of the product of the sides of an equilateral triangle to its perimeter is equal to the ratio of the product of the sides of another equilateral triangle to its perimeter . Then the triangles are

Answer

To solve the problem, we need to analyze the relationship between the sides of two equilateral triangles based on the given ratio of the product of their sides to their perimeter.

### Step-by-Step Solution:

1. **Understand the Properties of an Equilateral Triangle**:
- In an equilateral triangle, all three sides are equal. Let's denote the side length of the first triangle as \( a \) and the second triangle as \( b \).

2. **Calculate the Product of the Sides**:
- For the first triangle, the product of the sides is:
\[ \text{Product of sides} = a \times a \times a = a^3 \]
- For the second triangle, the product of the sides is:
\[ \text{Product of sides} = b \times b \times b = b^3 \]

3. **Calculate the Perimeter**:
- The perimeter of the first triangle is:
\[ \text{Perimeter} = a + a + a = 3a \]
- The perimeter of the second triangle is:
\[ \text{Perimeter} = b + b + b = 3b \]

4. **Set Up the Ratios**:
- According to the problem, the ratio of the product of the sides to the perimeter for both triangles is equal:
\[ \frac{a^3}{3a} = \frac{b^3}{3b} \]

5. **Simplify the Ratios**:
- Simplifying both sides gives:
\[ \frac{a^2}{3} = \frac{b^2}{3} \]
- We can eliminate the common factor of \( \frac{1}{3} \):
\[ a^2 = b^2 \]

6. **Conclude the Relationship Between the Sides**:
- From \( a^2 = b^2 \), we can conclude that:
\[ a = b \quad \text{or} \quad a = -b \]
- Since side lengths cannot be negative, we have:
\[ a = b \]

7. **Final Conclusion**:
- Since the sides of both triangles are equal, we conclude that the two triangles are **congruent**.