To find the centroid of triangle ABC, we use the formula for the centroid \( G \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \):
\[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
Given the vertices:
- \( A(-3, -3) \)
- \( B(0, 6) \)
- \( C(4, -3) \)
Substituting in the coordinates:
\[ G\left( \frac{-3 + 0 + 4}{3}, \frac{-3 + 6 - 3}{3} \right) \]
Calculating the x-coordinate of the centroid:
\[ \frac{-3 + 0 + 4}{3} = \frac{1}{3} \]
Calculating the y-coordinate of the centroid:
\[ \frac{-3 + 6 - 3}{3} = \frac{0}{3} = 0 \]
Thus, the coordinates of the centroid are:
\[ G\left( \frac{1}{3}, 0 \right) \]
This shows that all three medians intersect at the centroid, confirming that they meet at a single point.
Final Answer: \(\left( \frac{1}{3}, 0 \right)\)