Understanding the Centroid of a Triangle

The centroid of a triangle is the point where its three medians intersect. A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. The centroid is also known as the geometric center or the center of mass of the triangle.

Centroid Formula

Given the coordinates of the three vertices of a triangle A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃), the coordinates of the centroid (G) are calculated using the following formula:

G = ((x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3)

Step-by-Step Example

Let's find the centroid of a triangle with vertices A(1, 2), B(7, 8), and C(4, 5).

1. Identify the coordinates:
   • x₁ = 1, y₁ = 2
   • x₂ = 7, y₂ = 8
   • x₃ = 4, y₃ = 5
2. Apply the x-coordinate formula:
   • Gx = (x₁ + x₂ + x₃) / 3
   • Gx = (1 + 7 + 4) / 3
   • Gx = 12 / 3
   • Gx = 4
3. Apply the y-coordinate formula:
   • Gy = (y₁ + y₂ + y₃) / 3
   • Gy = (2 + 8 + 5) / 3
   • Gy = 15 / 3
   • Gy = 5
4. Combine the coordinates:
   • The centroid G is (4, 5).