Find the coor dinates of the centrod of a triangle whose coordinates are (3, -5, 7), (-1, 7, -6)
and (1, 1, 2).

Answer

To find the coordinates of the centroid of a triangle with given vertices, we can follow these steps:

### Step 1: Identify the coordinates of the vertices.
The vertices of the triangle are given as:
- Vertex A: \( (3, -5, 7) \)
- Vertex B: \( (-1, 7, -6) \)
- Vertex C: \( (1, 1, 2) \)

### Step 2: Use the formula for the centroid.
The formula for the centroid \( G(x, y, z) \) of a triangle with vertices \( (x_1, y_1, z_1) \), \( (x_2, y_2, z_2) \), and \( (x_3, y_3, z_3) \) is given by:
\[ G_x = \frac{x_1 + x_2 + x_3}{3}, \quad G_y = \frac{y_1 + y_2 + y_3}{3}, \quad G_z = \frac{z_1 + z_2 + z_3}{3} \]

### Step 3: Substitute the coordinates into the formula.
- For the x-coordinate:
\[ G_x = \frac{3 + (-1) + 1}{3} = \frac{3 - 1 + 1}{3} = \frac{3}{3} = 1 \]

- For the y-coordinate:
\[ G_y = \frac{-5 + 7 + 1}{3} = \frac{-5 + 8}{3} = \frac{3}{3} = 1 \]

- For the z-coordinate:
\[ G_z = \frac{7 + (-6) + 2}{3} = \frac{7 - 6 + 2}{3} = \frac{3}{3} = 1 \]

### Step 4: Combine the coordinates to find the centroid.
Thus, the coordinates of the centroid \( G \) are:
\[ G(1, 1, 1) \]

### Final Answer:
The coordinates of the centroid of the triangle are \( (1, 1, 1) \).

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