If the sides of an equilateral triangle be increased by 1 m its area is increased by `sqrt(3)` sq. metre. The length of any of its sides is

Options

A. 2 metre
B. `(5)/(2)` metre
C. `(3)/(2)` metre
D. `sqrt(3)`

Answer

To solve the problem, we need to find the length of the side of an equilateral triangle given that increasing the sides by 1 meter increases the area by \( \sqrt{3} \) square meters.

### Step-by-Step Solution:

1. **Define the Side Length**:
Let the length of each side of the equilateral triangle be \( a \) meters.

2. **Calculate the Area of the Original Triangle**:
The area \( A \) of an equilateral triangle is given by the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]

3. **Calculate the Area of the New Triangle**:
When the sides are increased by 1 meter, the new side length becomes \( a + 1 \). The area of the new triangle is:
\[ A' = \frac{\sqrt{3}}{4} (a + 1)^2 \]

4. **Set Up the Equation for the Increase in Area**:
According to the problem, the increase in area is \( \sqrt{3} \) square meters. Therefore, we can write:
\[ A' - A = \sqrt{3} \]
Substituting the areas we calculated:
\[ \frac{\sqrt{3}}{4} (a + 1)^2 - \frac{\sqrt{3}}{4} a^2 = \sqrt{3} \]

5. **Factor Out the Common Term**:
We can factor out \( \frac{\sqrt{3}}{4} \):
\[ \frac{\sqrt{3}}{4} \left( (a + 1)^2 - a^2 \right) = \sqrt{3} \]

6. **Simplify the Expression**:
Now simplify \( (a + 1)^2 - a^2 \):
\[ (a + 1)^2 - a^2 = a^2 + 2a + 1 - a^2 = 2a + 1 \]
So we have:
\[ \frac{\sqrt{3}}{4} (2a + 1) = \sqrt{3} \]

7. **Eliminate the Common Factor**:
To eliminate \( \sqrt{3} \), we can divide both sides by \( \sqrt{3} \):
\[ \frac{1}{4} (2a + 1) = 1 \]

8. **Multiply Both Sides by 4**:
Multiply both sides by 4 to get rid of the fraction:
\[ 2a + 1 = 4 \]

9. **Solve for \( a \)**:
Subtract 1 from both sides:
\[ 2a = 3 \]
Now divide by 2:
\[ a = \frac{3}{2} \]

10. **Final Answer**:
The length of any side of the triangle is:
\[ a = \frac{3}{2} \text{ meters} \]