Centroid of a Triangle

Centroid of a Triangle

How to solve the centroid of a triangle problems: definition, properties, examples, and their solutions.

Definition

The centroid of a triangle is the balance point of the triangle. The coordinates of the centriod are the average of each triangle's vertex.

The centroid of a triangle
is the balance point of the triangle.

By putting the centroid of the triangle
on the tip of a finger,
it'll be balanced.

So its coordinates
are the means of each triangle's vertex.

Example 1

Find the coordinates of the centriod of triangle ABC. A(3, 7), B(-2, 0), C(5, -4).

M(x value's mean, y value's mean)

Properties

Three medians of a triangle meet at the centriod.

Three medians of a triangle
meet at the centriod.

The centroid divides each median in the ratio of 2 : 1.

The centroid divides each median
in the ratio of 2 : 1.

(purple) : (blue) = 2 : 1

Example 2

Point M is the centroid of triangle ABC. Find the values of x and y. AM = 8, MP = 3x - 2, BP = 5y + 11, PC = 6.

Point M is the centroid.

So AP is the median of △ABC.

Then BP = PC.
5y + 11 = 6

Point M, the centroid, divides AP
in the ratio of 2 : 1.
(purple) : (blue) = 2 : 1

So 8 : 3x - 2 = 2 : 1.

Proportion