Business Mathematics and Basic Statistics · Ch 5 — Coordinate Geometry (Two Dimensions)

Centroid of a Triangle

A triangle’s median is the line segment joining one vertex to the mid-point of the opposite side; every triangle has three medians, one from each vertex. A classical geometric fact is that the three medians of any triangle always meet at a single point, called the centroid, and that the centroid divides each median in the ratio $2:1$ , measured from the vertex.

Because the centroid is such a specific point on each median (two-thirds of the way along it from the vertex), its coordinates can be obtained directly from the triangle’s three vertices, without drawing a single median, simply by averaging the corresponding coordinates.

Note

Centroid of a Triangle

If a triangle has vertices $A(x_1,y_1)$ , $B(x_2,y_2)$ and $C(x_3,y_3)$ , its centroid $G$ is

$G(x, y) = \left(\dfrac{x_1+x_2+x_3}{3},\ \dfrac{y_1+y_2+y_3}{3}\right)$

This averaging formula can, in fact, be derived from the section formula of the previous section: the centroid is precisely the point that divides a median (from a vertex to the mid-point of the opposite side) in the ratio $2:1$ from that vertex, and substituting the mid-point’s coordinates with this ratio into the section formula reproduces the boxed formula above. There is no need to redo this derivation for every problem — the boxed formula is applied directly.

Figure 4 — Triangle ABC with vertices A(2,3), B(4,7), C(6,-1); its three medians meet at the centroid G(4,3)

…

Start your 14-day free trial to unlock the full solution →

← Previous · 3Section Formula (Internal Division)Next →Example 1 — State the quadrant, or the axis, in which each of the following points…