Centroid of a Triangle

The centroid of a triangle is the point where all three medians of the triangle intersect with each other. It is one of the four classical centres of a triangle (centroid, orthocentre, circumcentre, and incentre). It represents the center of mass of a triangle and plays a key role in coordinate geometry and constructions. In this guide, you will learn the definition of the centroid of a triangle, the formula to find the centroid of a triangle, and solved problems for easy understanding.

Table of Contents

• What is the Centroid of a Triangle
• Solved Examples on the Centroid of a Triangle
• Practice Questions on the Centroid of a Triangle

What is the Centroid of a Triangle

The centroid of a triangle is the intersection point of all its medians. The centroid formula is simply the arithmetic mean of the coordinates of the triangle. The x-coordinate of the centroid is the arithmetic mean of the x-coordinates, and the y-coordinate is the arithmetic mean of the y-coordinates of the three vertices.
Properties of the centroid:

• The centroid divides each median in the ratio 2:1 from the vertex.

• The centroid always lies inside the triangle. This holds for every type of triangle, whether acute, obtuse, or right-angled. 

• The centroid is the centre of gravity (balance point) of the triangle.

• A triangle has exactly one centroid.

• The centroid divides the triangle into six smaller triangles of equal area.

Let A (x₁, y₁), B (x2, y₂), and C (x₃, y₃) be the vertices of the triangle ABC whose medians are AD, BE and CF. So, D, E, and F are, respectively, the midpoints of BC, AC and AB.

The centroid of the triangle = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)
The centroid G divides the median from vertex A to the midpoint D of BC in the ratio AG:GD = 2:1.

Solved Examples on the Centroid of a Triangle

Example1 : Find the centroid of the triangle with vertices A(2, 4), B(6, 2), and C(4, 8).
Solution: (x1, y1) = (2, 4), (x2, y2) = (6, 2), (x3, y3) = (4, 8)
X-coordinate of the centroid = (x1 + x2 + x3)/3 = (2 + 6 + 4)/3 = 4
Y-coordinate of the centroid = (y1 + y2 + y3)/3 = (4 + 2 + 8)/3 = 14/3
∴ The centroid of the triangle is (4, 14/3).

Example 2: Find the centroid of the triangle with vertices P(-1, 3), Q(5, -2), R(2, 7).
Solution: (x1, y1) = (-1, 3), (x2, y2) = (5, -2), (x3, y3) = (2, 7)
X-coordinate of the centroid = (x1 + x2 + x3)/3 = (-1 + 5 + 2)/3 = 2
Y-coordinate of the centroid = (y1 + y2 + y3)/3 = (3 + -2 + 7)/3 = 8/3
∴ The centroid of the triangle is (2, 8/3).

Example 3: The centroid of a triangle is G(3, 5). Two vertices are A(1, 7) and B(5, 3). Find the third vertex C.
Solution: The centroid of the triangle is (3, 5).Two vertices are A(1, 7) and B(5, 3).  Let the third vertex be C (a,b)
X-coordinate of the centroid = (x1 + x2 + x3)/3 = (1 + 5 + a)/3 = 3.
∴ a = 9 - 6 = 3
Y-coordinate of the centroid = (y1 + y2 + y3)/3 = ( 7 + 3 + b)/3 = 5
∴ b = 15 - 10 = 5.
Hence, the third vertex be C = (3, 5).

Practice Questions on the Centroid of a Triangle

1. The medians of a triangle meet at (2, 4). If two vertices are at the origin and (6, 0), find the third vertex.

2. If the centroid of a triangle with vertices (x, 2), (1, y), (5, -1) is (2, 3), find x and y.

3. Find the centroid of a right triangle with hypotenuse endpoints at (0, 8) and (6, 0) and the right angle at the origin.