Question

When the three medians of the triangle are drawn, they meet at a single point. What is the point of the centroid? Leave all results in fractions.

Answer 1

The centroid of a triangle is the point where the three medians intersect. The centroid is also known as the center of mass or barycenter of the triangle. It is found by taking the average of the triangle's vertices coordinates.

If the vertices of the triangle are given by the coordinates \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \), the coordinates of the centroid \( G \) can be calculated using the following formulas:

\[
G_x = \frac{x_1 + x_2 + x_3}{3}
\]

\[
G_y = \frac{y_1 + y_2 + y_3}{3}
\]

Thus, the coordinates of the centroid \( G \) can be expressed as:

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

In this way, the centroid divides each median in the ratio \( 2:1 \), with the longer segment being closer to the vertex. The centroid is always located inside the triangle regardless of the shape of the triangle.