Question

The three medians of △ABC meet at a single point. What is the point of the centroid? Express all results in fractions. (1 point) centroid= (,)
a(2,2)
B(8,2)
C(4,7)

Answer 1

To find the centroid of triangle \( ABC \) with vertices \( A(2,2) \), \( B(8,2) \), and \( C(4,7) \), we use the formula for the centroid \( G \) of a triangle defined by its vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \):

\[
G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
\]

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

- \( x_1 = 2 \)
- \( y_1 = 2 \)
- \( x_2 = 8 \)
- \( y_2 = 2 \)
- \( x_3 = 4 \)
- \( y_3 = 7 \)

Calculating the x-coordinate of the centroid:

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

Calculating the y-coordinate of the centroid:

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

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

\[
G\left( \frac{14}{3}, \frac{11}{3} \right)
\]

Thus, the answer is:

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