Given:
Each side of an equilateral triangle is increased by 2 cm.
The area of the triangle increases by 3√3 cm².

To find:
The length of each side of the triangle and its area.

Solution:
Let's assume the length of each side of the original equilateral triangle is 'x' cm.

Step 1: Calculating the new side length
When each side of the triangle is increased by 2 cm, the new side length becomes 'x + 2' cm.

Step 2: Calculating the new area
The area of an equilateral triangle is given by the formula:
Area = (√3/4) * (side length)²

The original area of the triangle is:
Original Area = (√3/4) * x²

The new area of the triangle is:
New Area = (√3/4) * (x + 2)²

According to the given information, the new area is increased by 3√3 cm² compared to the original area.
So, we can write the equation as:
New Area - Original Area = 3√3

Substituting the area formulas, we have:
(√3/4) * (x + 2)² - (√3/4) * x² = 3√3

Step 3: Solving the equation
Let's solve the equation to find the value of 'x'.

Expanding and simplifying the equation, we get:
(√3/4) * (x² + 4x + 4) - (√3/4) * x² = 3√3

Simplifying further:
(√3/4) * x² + (√3/4) * 4x + (√3/4) * 4 - (√3/4) * x² = 3√3

Cancelling out the like terms, we get:
(√3/4) * 4x + (√3/4) * 4 = 3√3

Simplifying:
(√3/4) * 4(x + 1) = 3√3

Dividing both sides by (√3/4) * 4, we get:
x + 1 = (3√3 * 4) / (√3 * 4)

Simplifying further:
x + 1 = 12 / 3

x + 1 = 4

Subtracting 1 from both sides, we get:
x = 3

Step 4: Calculating the side length and area
The length of each side of the equilateral triangle is 'x' cm, which is equal to 3 cm.
The new side length is 'x + 2' cm, which is equal to 5 cm.

The area of the original equilateral triangle is:
Original Area = (√3/4) * x² = (√3/4) * 3² = (√3/4) * 9 = 9√3/4 cm²

The area of the new equilateral triangle is:
New Area = (√3