To determine the coordinates of the centroid of a triangle, we can use the formula for the centroid (G) of a triangle with vertices [tex]\(X(x_1, y_1)\)[/tex], [tex]\(Y(x_2, y_2)\)[/tex], and [tex]\(Z(x_3, y_3)\)[/tex]:
[tex]\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \][/tex]
Given the vertices of the triangle:
- [tex]\(X(-4, 0)\)[/tex]
- [tex]\(Y(-1, 4)\)[/tex]
- [tex]\(Z(2, 2)\)[/tex]
We plug these coordinates into the centroid formula:
1. Calculate the x-coordinate of the centroid:
[tex]\[ \text{centroid}_x = \frac{-4 + (-1) + 2}{3} \][/tex]
2. Calculate the y-coordinate of the centroid:
[tex]\[ \text{centroid}_y = \frac{0 + 4 + 2}{3} \][/tex]
Lets perform these calculations step by step:
- For the x-coordinate:
[tex]\[ \text{centroid}_x = \frac{-4 - 1 + 2}{3} = \frac{-3}{3} = -1 \][/tex]
- For the y-coordinate:
[tex]\[ \text{centroid}_y = \frac{0 + 4 + 2}{3} = \frac{6}{3} = 2 \][/tex]
Therefore, the coordinates of the centroid of the triangle are:
[tex]\[ (-1, 2) \][/tex]
So, the final answer is:
[tex]\[\boxed{-1} \quad \boxed{2}\][/tex]
