Question

C (4,7)
B (8,2)
A (2,2)

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

centroid = ( _, _)

Answer 1

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:

1. Calculate the average of the x-coordinates:
\[
x_G = \frac{2 + 8 + 4}{3} = \frac{14}{3}
\]

2. 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:

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