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 of a triangle in three-dimensional space, we use the formula for the centroid \( G \) given the vertices \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \), and \( C(x_3, y_3, z_3) \):

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

Given the vertices of the triangle:
- \( A(1, 1, 0) \)
- \( B(1, 2, 1) \)
- \( C(-2, 2, -1) \)

We can identify the coordinates:
- \( x_1 = 1, y_1 = 1, z_1 = 0 \)
- \( x_2 = 1, y_2 = 2, z_2 = 1 \)
- \( x_3 = -2, y_3 = 2, z_3 = -1 \)

### Step 1: Calculate the x-coordinate of the centroid
\[ x_G = \frac{x_1 + x_2 + x_3}{3} = \frac{1 + 1 - 2}{3} = \frac{0}{3} = 0 \]

### Step 2: Calculate the y-coordinate of the centroid
\[ y_G = \frac{y_1 + y_2 + y_3}{3} = \frac{1 + 2 + 2}{3} = \frac{5}{3} \]

### Step 3: Calculate the z-coordinate of the centroid
\[ z_G = \frac{z_1 + z_2 + z_3}{3} = \frac{0 + 1 - 1}{3} = \frac{0}{3} = 0 \]

### Final Result
Thus, the coordinates of the centroid \( G \) are:
\[ G = \left( 0, \frac{5}{3}, 0 \right) \]

### Summary
The centroid of the triangle with vertices \( (1, 1, 0) \), \( (1, 2, 1) \), and \( (-2, 2, -1) \) is \( \left( 0, \frac{5}{3}, 0 \right) \).