To find the centroid of triangle ABC, we can use the formula for the centroid, which is given by the average of the coordinates of the vertices of the triangle. The vertices of triangle ABC are:
- \( A(-3, -3) \)
- \( B(0, 6) \)
- \( C(4, -3) \)
The coordinates of the centroid \( G \) can be calculated using the following formula:
\[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
Substituting in the coordinates of points A, B, and C:
\[ G\left( \frac{-3 + 0 + 4}{3}, \frac{-3 + 6 - 3}{3} \right) \]
Calculating the x-coordinate of the centroid:
\[ G_x = \frac{-3 + 0 + 4}{3} = \frac{1}{3} \]
Calculating the y-coordinate of the centroid:
\[ G_y = \frac{-3 + 6 - 3}{3} = \frac{0}{3} = 0 \]
Thus, the coordinates of the centroid \( G \) are:
\[ G\left( \frac{1}{3}, 0 \right) \]
To confirm that the medians intersect at the centroid, we can see that the centroid divides each median in a ratio of \( 2:1 \). Since we have determined the coordinates of the centroid, we can conclude that the medians indeed intersect at this single point.
Thus, we have proven that the medians of triangle ABC meet at a single point, the centroid \( G\left( \frac{1}{3}, 0 \right) \).