To find the centroid of triangle ABC with vertices A(-3, -3), B(0, 6), and C(4, -3), we can use the formula for the centroid, which is given by:
\[ G\left( \frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3} \right) \]
where \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are the coordinates of the vertices of the triangle.
Let's identify the coordinates of each vertex:
- \(A = (-3, -3)\)
- \(B = (0, 6)\)
- \(C = (4, -3)\)
Next, we substitute the coordinates into the centroid formula:
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Calculate the \(x\)-coordinate of the centroid: \[ x_G = \frac{-3 + 0 + 4}{3} = \frac{1}{3} \]
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Calculate the \(y\)-coordinate of the centroid: \[ y_G = \frac{-3 + 6 - 3}{3} = \frac{0}{3} = 0 \]
Therefore, the coordinates of the centroid \(G\) of triangle ABC are: \[ G\left( \frac{1}{3}, 0 \right) \]
Now we have shown that the centroid (the point where the medians meet) is at the coordinates \(\left( \frac{1}{3}, 0 \right)\). This proves that the medians of triangle ABC meet at a single point, which is the centroid \(G\).
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