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=
(
,
)

Answer 1

The centroid of a triangle \( \triangle ABC \) is the point where the three medians intersect. It can be calculated using the coordinates of the vertices of the triangle.

If the vertices of \( \triangle ABC \) are given by the coordinates:
- \( A(x_1, y_1) \)
- \( B(x_2, y_2) \)
- \( C(x_3, y_3) \)

The centroid \( G \) can be found using the formula:

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

To express this as a fraction, you can use the concept that each coordinate of the centroid is the average of the coordinates of the vertices.

For instance, if we have points:
- A(1, 2)
- B(3, 4)
- C(5, 6)

The centroid \( G \) would be calculated as follows:

\[
G_x = \frac{1 + 3 + 5}{3} = \frac{9}{3} = 3
\]
\[
G_y = \frac{2 + 4 + 6}{3} = \frac{12}{3} = 4
\]

Thus, the centroid \( G \) would be:

\[
G = (3, 4)
\]

If points \( A \), \( B \), and \( C \) are not specified, please provide them, and we can give you the specific centroid. Otherwise, the general formula for the centroid of \( \triangle ABC \) is:

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