To find the centroid of triangle ABC, we need to calculate the average of the x-coordinates and the average of the y-coordinates of the vertices A, B, and C.
The coordinates of the points are:
- A(2, 2)
- B(8, 2)
- C(4, 7)
The formula for the centroid \( G \) of a triangle with vertices at \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
Substituting the coordinates of points A, B, and C into the formula:
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Calculate the average of the x-coordinates: \[ x_G = \frac{2 + 8 + 4}{3} = \frac{14}{3} \]
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Calculate the average of the y-coordinates: \[ y_G = \frac{2 + 2 + 7}{3} = \frac{11}{3} \]
Thus, the coordinates of the centroid \( G \) are:
\[ G = \left( \frac{14}{3}, \frac{11}{3} \right) \]
Therefore, the final answer is:
\[ \text{centroid} = \left( \frac{14}{3}, \frac{11}{3} \right) \]
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