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(x1, y1), B(x2, y2), C(x3, y3),
the centroid is
M([x1 + x2 + x3]/3, [y1 + y2 + y3]/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.