If side of an equilateral triangle is increased by 2 units , then the area is increased by `3+sqrt3` square units. Find the side of this equilateral triangle.

Answer

To find the side of the equilateral triangle given that increasing its side by 2 units increases the area by \(3 + \sqrt{3}\) square units, we can follow these steps:

### Step 1: Define the side of the equilateral triangle
Let the side of the equilateral triangle be \(A\).

### Step 2: Write the formula for the area of the equilateral triangle
The area \(A_{old}\) of an equilateral triangle with side \(A\) is given by the formula:
\[ A_{old} = \frac{\sqrt{3}}{4} A^2 \]

### Step 3: Write the formula for the new area after increasing the side
When the side is increased by 2 units, the new side becomes \(A + 2\). The area \(A_{new}\) of the triangle with the new side is:
\[ A_{new} = \frac{\sqrt{3}}{4} (A + 2)^2 \]

### Step 4: Set up the equation for the increase in area
According to the problem, the increase in area is given by:
\[ A_{new} - A_{old} = 3 + \sqrt{3} \]
Substituting the expressions for \(A_{new}\) and \(A_{old}\):
\[ \frac{\sqrt{3}}{4} (A + 2)^2 - \frac{\sqrt{3}}{4} A^2 = 3 + \sqrt{3} \]

### Step 5: Simplify the equation
Factor out \(\frac{\sqrt{3}}{4}\):
\[ \frac{\sqrt{3}}{4} \left((A + 2)^2 - A^2\right) = 3 + \sqrt{3} \]
Now, expand \((A + 2)^2\):
\[ (A + 2)^2 = A^2 + 4A + 4 \]
So,
\[ \frac{\sqrt{3}}{4} \left(A^2 + 4A + 4 - A^2\right) = 3 + \sqrt{3} \]
This simplifies to:
\[ \frac{\sqrt{3}}{4} (4A + 4) = 3 + \sqrt{3} \]

### Step 6: Multiply both sides by 4 to eliminate the fraction
\[ \sqrt{3} (4A + 4) = 12 + 4\sqrt{3} \]

### Step 7: Distribute \(\sqrt{3}\)
\[ 4\sqrt{3} A + 4\sqrt{3} = 12 + 4\sqrt{3} \]

### Step 8: Isolate the term with \(A\)
Subtract \(4\sqrt{3}\) from both sides:
\[ 4\sqrt{3} A = 12 \]

### Step 9: Solve for \(A\)
Divide both sides by \(4\sqrt{3}\):
\[ A = \frac{12}{4\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \]

### Conclusion
The side of the equilateral triangle is:
\[ \boxed{\sqrt{3}} \]