# Triangle: Centroid

How to find the centroid of a triangle and use the property of the centroid: formula, 2 properties, 2 examples, and their solutions.

## Formula

### Formula

For a triangle whose points are

A(x_{1}, y_{1}), B(x_{2}, y_{2}), C(x_{3}, y_{3}),

the centroid is

M([x_{1} + x_{2} + x_{3}]/3, [y_{1} + y_{2} + y_{3}]/3).

The coordinates of the centroid

is the mean of the points of a triangle.

## Example 1

### Example

### Solution

The points of the triangle are

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

Then the centroid M is,

the mean of the x values, [3 + (-2) + 5]/3

comma,

the mean of the y values, [7 + 0 + (-4)]/3.

3 + (-2) = 3 - 2 = 1

7 + 0 + (-4) = 7 - 4 = 3

1 + 5 = 6

3/3 = 1

6/3 = 2

So the centroid is (2, 1).

## Property

The centroid of a triangle has two properties.

### Property 1

Three medians of a triangle

meet at the centroid.

### Property 2

The centroid divides each median

in the ratio of 2 : 1.

So for each median,

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

## Example 2

### Example

### Solution

AP passes through the centroid M.

So AP is the median of △ABC.

So BP = PC.

BP = 5y + 11

PC = 6

So 5y + 11 = 6.

Move +11 to the right side.

Then 5y = -5.

Divide both sides by 5.

Then y = -1.

Next, see AP.

AP is the median.

And M is the centroid.

AM = 8

MP = 3x - 2

So

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

Solve the proportion.

Then 2(3x - 2) = 8.

Divide both sides by 2.

Then 3x - 2 = 4.

Move -2 to the right side.

Then 3x = 6.

Divide both sides by 3.

Then x = 2.

y = -1

x = 2

So write x = 2 and y = 1.

So x = 2 and y = 1 is the answer.