there is a triangle made with 3 points, A (2,2), B(8,2), and C(4,7)

Question

The three medians of ABC meet at a single point. What is the point of the centroid? Express all results in fractions.

centroid=(_,_)

Answer 1

The centroid of a triangle is the average of the coordinates of its vertices. Given the points \( A(2, 2) \), \( B(8, 2) \), and \( C(4, 7) \), the centroid \( G \) can be calculated using the following formula:

\[ G_x = \frac{x_1 + x_2 + x_3}{3} \] \[ G_y = \frac{y_1 + y_2 + y_3}{3} \]

Substituting the coordinates of points \( A \), \( B \), and \( C \):

\[ G_x = \frac{2 + 8 + 4}{3} = \frac{14}{3} \]

\[ G_y = \frac{2 + 2 + 7}{3} = \frac{11}{3} \]

Thus, the centroid \( G \) of triangle \( ABC \) is:

\[ \text{centroid} = \left( \frac{14}{3}, \frac{11}{3} \right) \]

So the final result is:

\[ \text{centroid} = \left( \frac{14}{3}, \frac{11}{3} \right) \]