The centroid of the triangle with vertices `(1, sqrt(3)), (0, 0)` and (2, 0) is

Options

A. `(1, (sqrt(3))/(2))`
B. `((2)/(3), (1)/(sqrt(3)))`
C. `((2)/(3), (sqrt(3))/(2))`
D. `(1, (1)/(sqrt(3)))`

Answer

To find the centroid of the triangle with vertices \( A(1, \sqrt{3}) \), \( B(0, 0) \), and \( C(2, 0) \), we will use the formula for the centroid of a triangle given by the coordinates of its vertices. The formula for the centroid \( G(x, y) \) is:

\[ G\left(x, y\right) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

### Step 1: Identify the coordinates of the vertices
The coordinates of the vertices are:
- \( A(1, \sqrt{3}) \) where \( x_1 = 1 \) and \( y_1 = \sqrt{3} \)
- \( B(0, 0) \) where \( x_2 = 0 \) and \( y_2 = 0 \)
- \( C(2, 0) \) where \( x_3 = 2 \) and \( y_3 = 0 \)

### Step 2: Substitute the coordinates into the centroid formula
Now we substitute the coordinates into the centroid formula:

\[ G\left(x, y\right) = \left(\frac{1 + 0 + 2}{3}, \frac{\sqrt{3} + 0 + 0}{3}\right) \]

### Step 3: Calculate the x-coordinate of the centroid
Calculating the x-coordinate:

\[ x = \frac{1 + 0 + 2}{3} = \frac{3}{3} = 1 \]

### Step 4: Calculate the y-coordinate of the centroid
Calculating the y-coordinate:

\[ y = \frac{\sqrt{3} + 0 + 0}{3} = \frac{\sqrt{3}}{3} \]

### Step 5: Final coordinates of the centroid
Thus, the coordinates of the centroid \( G \) are:

\[ G\left(1, \frac{\sqrt{3}}{3}\right) \]

### Step 6: Rationalize the y-coordinate
To express the y-coordinate in a different form, we can rationalize it:

\[ \frac{\sqrt{3}}{3} = \frac{1}{\sqrt{3}} \]

So the coordinates of the centroid can also be expressed as:

\[ G\left(1, \frac{1}{\sqrt{3}}\right) \]

### Conclusion
The centroid of the triangle with vertices \( (1, \sqrt{3}), (0, 0), (2, 0) \) is:

\[ \boxed{\left(1, \frac{1}{\sqrt{3}}\right)} \]