C (4,7)

Question

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:

$$G=(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$$

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:

$$G=(\frac{14}{3},\frac{11}{3})$$

Therefore, the final answer is:

$$\text{centroid}=(\frac{14}{3},\frac{11}{3})$$