In the `DeltaABC`, the coordinates of B are `(0, 0), AB=2, /_ABC=pi/3` and the middle point of BC has the coordinates (2,0). The centroid of the triangle is

Options

A. `(1//2,sqrt3//2)`
B. `(5//3,1//sqrt3)`
C. `(4+sqrt3//3,1//3)`
D. none of these

Explanation

To find the centroid of triangle ABC given the coordinates of point B, the length of AB, and the angle ABC, we can follow these steps:

### Step 1: Identify the coordinates of point B
Given that the coordinates of point B are (0, 0), we can denote:
- \( B(0, 0) \)

### Step 2: Determine the coordinates of point C
We know that the midpoint of BC is (2, 0). Let the coordinates of point C be \( C(x_C, y_C) \). Since the midpoint M of segment BC is given by:
\[ M = \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2} \right) \]
Substituting the known values:
\[ (2, 0) = \left( \frac{0 + x_C}{2}, \frac{0 + y_C}{2} \right) \]
From the x-coordinate:
\[ 2 = \frac{x_C}{2} \implies x_C = 4 \]
From the y-coordinate:
\[ 0 = \frac{y_C}{2} \implies y_C = 0 \]
Thus, the coordinates of point C are:
- \( C(4, 0) \)

### Step 3: Determine the coordinates of point A
We are given that \( AB = 2 \) and \( \angle ABC = \frac{\pi}{3} \) (which is 60 degrees). The coordinates of point A can be found using polar coordinates:
\[ A(x_A, y_A) = (r \cos \theta, r \sin \theta) \]
where \( r = 2 \) and \( \theta = \frac{\pi}{3} \):
\[ x_A = 2 \cos\left(\frac{\pi}{3}\right) = 2 \cdot \frac{1}{2} = 1 \]
\[ y_A = 2 \sin\left(\frac{\pi}{3}\right) = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \]
Thus, the coordinates of point A are:
- \( A(1, \sqrt{3}) \)

### Step 4: Calculate the centroid of triangle ABC
The centroid \( G \) of triangle ABC is given by the formula:
\[ G\left( \frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3} \right) \]
Substituting the coordinates of points A, B, and C:
\[ G\left( \frac{1 + 0 + 4}{3}, \frac{\sqrt{3} + 0 + 0}{3} \right) = G\left( \frac{5}{3}, \frac{\sqrt{3}}{3} \right) \]

### Final Answer
The coordinates of the centroid \( G \) are:
\[ G\left( \frac{5}{3}, \frac{\sqrt{3}}{3} \right) \]