If the vertices of a triangle be (1, 1, 0), (1, 2, 1) and
`(-2, 2, -1),` the the centroid of the triangie is

Options

A. `(0, 5/3, 0)
B. (5, 0, 3)
C. (0, 0, 0)
D. (1, 2, 3)

Answer

To find the centroid (G) of the triangle with vertices A(1, 1, 0), B(1, 2, 1), and C(-2, 2, -1), we can use the formula for the centroid of a triangle in three-dimensional space:

\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \]

Where:
- \( (x_1, y_1, z_1) \) are the coordinates of vertex A,
- \( (x_2, y_2, z_2) \) are the coordinates of vertex B,
- \( (x_3, y_3, z_3) \) are the coordinates of vertex C.

### Step 1: Identify the coordinates of the vertices
- A = (1, 1, 0)
- B = (1, 2, 1)
- C = (-2, 2, -1)

### Step 2: Substitute the coordinates into the centroid formula
\[ G = \left( \frac{1 + 1 - 2}{3}, \frac{1 + 2 + 2}{3}, \frac{0 + 1 - 1}{3} \right) \]

### Step 3: Calculate the x-coordinate of the centroid
\[ x_G = \frac{1 + 1 - 2}{3} = \frac{0}{3} = 0 \]

### Step 4: Calculate the y-coordinate of the centroid
\[ y_G = \frac{1 + 2 + 2}{3} = \frac{5}{3} \]

### Step 5: Calculate the z-coordinate of the centroid
\[ z_G = \frac{0 + 1 - 1}{3} = \frac{0}{3} = 0 \]

### Step 6: Combine the coordinates to find the centroid
Thus, the centroid G of the triangle is:
\[ G = \left( 0, \frac{5}{3}, 0 \right) \]

### Final Answer
The centroid of the triangle is \( G = \left( 0, \frac{5}{3}, 0 \right) \).