Centroid of a Triangle

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. 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 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

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

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

Three medians of a triangle meet at the centriod.

Property 1:

Three medians of a triangle
meet at the centroid.

Medians of a triangle - Definition

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

Property 2:

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

So for each median,
[blue] : [green] = 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.

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.

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 = 3x - 2

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

Solve the proportion.

Then 2(3x - 2) = 8.

Proportion

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.

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