To find the coordinates of the centroid of a triangle with given vertices [tex]\( A(-6, 0) \)[/tex], [tex]\( B(-4, 4) \)[/tex], and [tex]\( C(0, 2) \)[/tex], you use the centroid formula.
The centroid [tex]\( G \)[/tex] of a triangle can be found by taking the average of the x-coordinates and the average of the y-coordinates of the vertices.
1. Calculate the x-coordinate of the centroid:
[tex]\[
x_G = \frac{x_A + x_B + x_C}{3}
\][/tex]
Plugging in the x-coordinates of the vertices:
[tex]\[
x_G = \frac{-6 + (-4) + 0}{3} = \frac{-6 - 4 + 0}{3} = \frac{-10}{3} = -\frac{10}{3} \approx -3.333
\][/tex]
2. Calculate the y-coordinate of the centroid:
[tex]\[
y_G = \frac{y_A + y_B + y_C}{3}
\][/tex]
Plugging in the y-coordinates of the vertices:
[tex]\[
y_G = \frac{0 + 4 + 2}{3} = \frac{0 + 4 + 2}{3} = \frac{6}{3} = 2
\][/tex]
Therefore, the coordinates of the centroid [tex]\( G \)[/tex] are:
[tex]\[
\left( -\frac{10}{3}, 2 \right) \approx (-3.333, 2)
\][/tex]
So, the coordinates of the centroid are:
[tex]\[
\boxed{-3.3333333333333335} \quad \boxed{2.0}
\][/tex]
