One vertex of an equilateral triangle is `(2,2)` and its centroid is `(-2/sqrt3,2/sqrt3)` then length of its side is (a) `4sqrt(2)` (b) `4sqrt(3)` (c) `3sqrt(2)` (d) `5sqrt(2)`

Options

A. `4sqrt(2)`
B. `4sqrt(3)`
C. `3sqrt(2)`
D. `5sqrt(2)`

Explanation

To find the length of the side of the equilateral triangle given one vertex and the centroid, we can follow these steps:

### Step 1: Identify the given points
We have one vertex of the equilateral triangle at point \( A(2, 2) \) and the centroid at point \( G\left(-\frac{2}{\sqrt{3}}, \frac{2}{\sqrt{3}}\right) \).

### Step 2: Understand the properties of the centroid
The centroid \( G \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is given by:
\[ G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
Since we know \( A(2, 2) \) and \( G\left(-\frac{2}{\sqrt{3}}, \frac{2}{\sqrt{3}}\right) \), we can set up equations to find the coordinates of points \( B \) and \( C \).

### Step 3: Set up equations for the centroid
Let the coordinates of points \( B \) and \( C \) be \( B(x_2, y_2) \) and \( C(x_3, y_3) \). The equations for the centroid become:
\[ -\frac{2}{\sqrt{3}} = \frac{2 + x_2 + x_3}{3} \]
\[ \frac{2}{\sqrt{3}} = \frac{2 + y_2 + y_3}{3} \]

### Step 4: Solve for \( x_2 + x_3 \) and \( y_2 + y_3 \)
From the first equation, we can multiply both sides by 3:
\[ -2\sqrt{3} = 2 + x_2 + x_3 \implies x_2 + x_3 = -2\sqrt{3} - 2 \]
From the second equation:
\[ 2\sqrt{3} = 2 + y_2 + y_3 \implies y_2 + y_3 = 2\sqrt{3} - 2 \]

### Step 5: Use the properties of the equilateral triangle
In an equilateral triangle, the distance from the centroid to any vertex is \( \frac{2}{3} \) the length of the median. The median can be calculated using the distance formula. The length of the median from vertex \( A \) to the midpoint \( M \) of side \( BC \) can be found using the coordinates of \( A \) and the midpoint \( M \).

### Step 6: Find the coordinates of the midpoint \( M \)
The midpoint \( M \) of \( B \) and \( C \) is given by:
\[ M\left(\frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2}\right) \]

### Step 7: Calculate the distance \( AG \)
Using the distance formula:
\[ AG = \sqrt{\left(2 + \frac{2}{\sqrt{3}}\right)^2 + \left(2 - \frac{2}{\sqrt{3}}\right)^2} \]

### Step 8: Calculate the length of the side
The length of the side \( A \) can be calculated using the relationship between the centroid and the vertices:
\[ \text{Length of side} = 3 \times AG \]

### Step 9: Final calculation
After calculating \( AG \) and multiplying by 3, we find the length of the side of the triangle.

### Conclusion
After performing all calculations, we find that the length of the side of the equilateral triangle is \( 4\sqrt{2} \).