# Centroid of a Triangle

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

## Definition

The centroid of a triangle

is the balance point of the triangle.

By putting the tip of a finger

right under the centroid,

it'll be balanced.

So the centroid *M* is

( [mean of the *x* values], [mean of the *y* values] ).

Mean (Average)

## Example 1

The vertices of △*ABC* are

(3, 7), (-2, 0), and (5, -4).

So the centroid *M* is,

the mean of the *x* values,

[3 + (-2) + 5]/3,

the mean of the *y* values,

[7 + 0 + (-4)]/3.

Mean (Average)

3 + (-2) = 1

7 + 0 + (-4) = 3

1 + 5 = 6

3/3 = 1

6/3 = 2

So the centroid *M* is (2, 1).

## Properties

Property 1:

Three medians of a triangle

meet at the centroid.

Medians of a triangle - Definition

Property 2:

The centroid divides each median

in the ratio of 2 : 1.

So for each median,

[blue] : [green] = 2 : 1.

## Example 2

*AP* passes through the centroid *M*.

So *AP* is the median of △*ABC*.

So *BP* = *PC*.*BP* = 5*y* + 11*PC* = 6

So 5*y* + 11 = 6.

Move +11 to the right side.

Then 5*y* = -5.

Divide both sides by 5.

Then *y* = -1.

*AP* passes through the centroid *M*.

So the centroid *M* divides *AP*

in the ratio of 2 : 1.

So *AM* : *MP* = 2 : 1.*AM* = 8*MP* = 3*x* - 2

So 8 : (3*x* - 2) = 2 : 1.

Solve the proportion.

Then 2(3*x* - 2) = 8.

Proportion

Divide both sides by 2.

Then 3*x* - 2 = 4.

Move -2 to the right side.

Then 3*x* = 6.

Divide both sides by 3.

Then *x* = 2.

so *x* = 2, *y* = -1 is the answer.