To find the centroid of triangle ABC with vertices A(-3, -3), B(0, 6), and C(4, -3), we use the formula for the centroid of a triangle, which is:
\[ G\left(x_G, y_G\right) = \left(\frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3}\right) \]
where \((x_A, y_A)\), \((x_B, y_B)\), and \((x_C, y_C)\) are the coordinates of the vertices of the triangle.
Plugging in the coordinates of the vertices:
- \(A(-3, -3)\)
- \(B(0, 6)\)
- \(C(4, -3)\)
We find the x-coordinate of the centroid \(x_G\):
\[ x_G = \frac{-3 + 0 + 4}{3} = \frac{1}{3} \]
Next, we find the y-coordinate of the centroid \(y_G\):
\[ y_G = \frac{-3 + 6 - 3}{3} = \frac{0}{3} = 0 \]
Thus, the coordinates of the centroid \(G\) are:
\[ G\left(\frac{1}{3}, 0\right) \]
Now, let's compare this result with the options given:
A. \((0, \frac{1}{3})\)
B. \((\frac{1}{6}, 3)\)
C. \((\frac{1}{3}, 0)\)
D. \((\frac{1}{3}, 1)\)
The correct answer is:
C. \((\frac{1}{3}, 0)\)
This proves that the medians of triangle ABC meet at a single point, which is the centroid \(G\).