To prove that the medians of triangle ABC meet at a single point, we can find the centroid (G) of the triangle. The centroid is the point where the three medians intersect, and it can be calculated using the coordinates of the vertices of the triangle.
Given the vertices:
- \( A(-3, -3) \)
- \( B(0, 6) \)
- \( C(4, -3) \)
The formula for the centroid \( G(x, y) \) of a triangle with vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is given by:
\[ G(x, y) = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
Now, substituting the coordinates of the vertices A, B, and C into this formula:
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The x-coordinate of the centroid: \[ x_G = \frac{-3 + 0 + 4}{3} = \frac{1}{3} \]
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The y-coordinate of the centroid: \[ y_G = \frac{-3 + 6 - 3}{3} = \frac{0}{3} = 0 \]
So the coordinates of the centroid \( G \) are: \[ G\left( \frac{1}{3}, 0 \right) \]
Thus, we have found that the centroid \( G \) of triangle ABC is located at \( \left( \frac{1}{3}, 0 \right) \).
Since the centroid divides each median in a 2:1 ratio and all three medians meet at this point, we can conclude that the medians of triangle ABC indeed meet at a single point, the centroid \( G \).
This completes our proof.