Question

Use the image to answer the question.

A scalene triangle made up of vertices upper A, upper B, and upper C, is plotted on a grid. Each of its vertices has their coordinates labeled, and they are connected to an interior point labeled upper D. A 2,2 B.8,2 C.4,7

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=
(
,
)

Answer 1

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

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

Substituting the coordinates of the vertices:

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

Calculating the \( x \)-coordinate of the centroid:

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

Calculating the \( y \)-coordinate of the centroid:

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

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

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

In conclusion, the centroid is

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