The centroid $G$ of a triangle with vertices $A(x_1, y_1, z_1)$, $B(x_2, y_2, z_2)$, and $C(x_3, y_3, z_3)$ is calculated using the formula:

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

Given:

• $A = (2, 2, 3)$
• $B = (x, 4, -1)$
• $C = (3, x, -5)$
• $G = (2, 1, -1)$

We equate the components of $G$ with the formula for the centroid.

$x$-coordinate:

$\frac{2 + x + 3}{3} = 2$ $2 + x + 3 = 6 \quad \text{(Multiply through by 3)}$ $x + 5 = 6$ $x = 1$

$y$-coordinate:

$\frac{2 + 4 + x}{3} = 1$ $2 + 4 + x = 3 \quad \text{(Multiply through by 3)}$ $6 + x = 3$ $x = -3$

This creates a conflict in $x$-values, which suggests a problem with the provided data or calculation. Please double-check the vertex coordinates.

Would you like help verifying this or revising the input? Meanwhile, here are related questions:

1. How is the centroid of a triangle defined in $n$-dimensional space?
2. What happens if all vertices are collinear?
3. Can you calculate the area of $\triangle ABC$ given its vertices?
4. How do negative coordinates affect centroid location?
5. How can we generalize centroid properties to non-triangular polygons?

Tip: Always verify all vertices are distinct before applying centroid formulas.