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Centroid of a Triangle

Centroid of a triangle is the point where the three medians of the triangle meet (We know that the line segment meeting the vertex and midpoint of opposite side is called its median). Centroid is also known as center of triangle. Following figure given below demonstrates centroid of a triangle : In above figure, point P is the centroid of $\bigtriangleup$ABC.

Centroid is also called center of mass. Center of mass of any geometrical figure is a point where whole mass of the body concentrates. Therefore, centroid of a triangle is the point at which its overall mass is focused. If we want to balance a triangle at the tip of pencil, we shall be able to do it very easily if we put the tip over the centroid of the triangle.

The centroid of the triangle divides the medians of a triangle in the ratio of 2 : 1 from the vertex as shown in the following figure : Thus, AG : GD = 2 : 1$\frac{AG}{GD}=\frac{2}{1}$

Formula For the Centroid of a Triangle

The position of centroid of a triangle can be easily found if we know the exact position of the three vertices of triangle in Cartesian plane. Let us consider a triangle $\bigtriangleup$ABC whose vertices have coordinates ($x_{1}, y_{1}$), (
$x_{2}, y_{2}$) and ($x_{3}, y_{3}$) as shown in diagram below : If the coordinates of centroid of that triangle are assumed to be (x, y), then -

x = $\frac{x_{1}+x_{2}+x_{3}}{3}$  and  y = $\frac{y_{1}+y_{2}+y_{3}}{3}$
We can say that the coordinates of centroid of a triangle can be calculated by estimating the average of x-coordinates and y-coordinates of three vertices separately.

Example:
Find the coordinates of centroid of the triangle whose vertices are positioned at (2, 1), B(5, 1) and C(4, 6).

Solution:
We have
$x_{1}$ = 2 ,
$y_{1}$ = 1
$x_{2}$ = 5, $y_{2}$ = 1
$x_{3}$ = 4, $y_{3}$ = 6
Centroid is given by :

x = $\frac{x_{1}+x_{2}+x_{3}}{3}$ and y = $\frac{y_{1}+y_{2}+y_{3}}{3}$

x = $\frac{2+5+4}{3}$ and y = $\frac{1+1+6}{3}$

x = $\frac{11}{3}$ and y = $\frac{8}{3}$

x = 3.67 and y = 2.67

Therefore, the coordinates of centroid are (3.67, 2.67).

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