Centroid of a Triangle

The point of intersection of the three medians of a triangle is called the centroid of a triangle.

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has exactly three medians, one from each vertex, and they all intersect each other at the triangle's centroid. 

(Image will be updated soon)

In the above figure, D is the midpoint of side BC, which divides BC into two halves i.e. BD = CD. and the line segment from vertex A joins it. So, AD is the median of ∆ ABC. Similarly, BE and CF are the medians from vertex B and C to their opposite side AC and AB, and bisect them respectively. The three medians AD, BE and CF intersect each other at a point G, the centroid of the triangle.

The Centroid of a triangle, also known as the geometrical centre of a triangle, is the point of intersection of the three medians of a triangle. A median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side, thereby bisecting that side. All triangles have exactly three medians, one from each vertex, and all of them intersect each other at the triangle's centroid.

Difference between a Median and Centroid

A median divides the triangle into two triangles which are of equal area. The centroid is known as the intersection of the three medians. The three medians divide the triangle into six triangles, and each of these six triangles has the same area. The centroid divides each median into two parts, and this division is always in the ratio of  2:1.

Properties of the Centroid of a Triangle

1. The centroid of a triangle is formed by the intersection of the medians of the triangle.

2. Medians always lie within a triangle. Therefore, their intersection point i.e., the centroid of a triangle always lies inside a triangle.

3. The centroid is usually denoted by ‘G’.

4. The centroid internally divides all the three medians in the ratio 2:1.

5. The Centroid theorem states that the centroid of a triangle is at a 23 distance from the vertex of a triangle and a distance of 13 from the side opposite to the vertex. 

In the above figure, AG = \[\frac{2}{3}\]AD and DG = \[\frac{1}{3}\]AD 

The Formula of Centroid of a Triangle

If the coordinates of the three vertices of a triangle are A (x1, y1), B (x2, y2), C (x3, y3), then the formula for the centroid of the triangle is given as below:

The centroid of a triangle G(x,y) = ((\[\frac{x_{1}+x_{2}+x_{3}}{3}\]) , (\[\frac{y_{1}+y_{2}+y_{3}}{3}\]))

Where,

x1, x2, x3 are the x coordinates of the vertices of a triangle.

y1, y2, y3 are the y coordinates of the vertices of a triangle.

Derivation of Formula of Centroid of a Triangle

Let ABC be a triangle with the vertex coordinates A (x1, y1), B (x2, y2), and C (x3, y3). The midpoints of the side BC, AC and AB are D, E, and F respectively. The centroid of a triangle is denoted as G. (the image will be updated soon)

As D is the midpoint of the side BC, Using the midpoint formula the coordinates of midpoint D can be calculated as:

D ((\[\frac{x_{2}+x_{3}}{3}\] , (\[\frac{y_{2}+y_{3}}{3}\]))

We know that centroid ‘G’ divides the medians in the ratio of 2: 1. Therefore, the coordinates of the centroid G (x, y) are calculated using the section formula:

To Find the X-Coordinates of G:

x =  \[\frac{2(\frac{x_{2}+x_{3}}{2})+1(x_{1})}{2+1}\]

x = (\[\frac{x_{1}+x_{2}+x_{3}}{3}\])

To Find the Y-Coordinates of G:

Similarly, To y-coordinates of the centroid ‘G’.

y = \[\frac{2(\frac{y_{2}+y_{3}}{2})+1(y_{1})}{2+1}\]

y = (\[\frac{y_{1}+y_{2}+y_{3}}{3}\])

Therefore, the coordinates of the centroid G (x, y) is ((\[\frac{x_{1}+x_{2}+x_{3}}{3}\]) , (\[\frac{x_{1}+x_{2}+x_{3}}{3}\]))

Solved Examples

1. Find the Coordinates of the Centroid of a Triangle ABC whose Vertices are A (1, 2), B (3, 4) and C (5, 6).

Ans: The coordinates of vertices of a triangle have been given as A (1, 2), B (3, 4) and C (5, 6).

On separating the x-coordinates of given vertices, we obtain:

x1 = 1, x2 = 3 and x3 = 5    

Similarly, for the y-coordinates:

y1 = 2, y2 = 4 and y3 = 6    

And the coordinates of the centroid G (x, y) of triangle ABC is given by

the x-coordinates of G:

And the coordinates of the centroid G (x, y) of triangle ABC is given by

the x-coordinates of G:

x = (\[\frac{x_{1}+x_{2}+x_{3}}{3}\]) 

On substituting the corresponding values of x1, x2, x3 in the above formula, we get:

x = (\[\frac{1+3+5}{3}\])

x = \[\frac{9}{3}\] 

x = 3.

the y-coordinates of G:

y = (\[\frac{y_{2}+y_{3}}{3}\])

On substituting the corresponding values of y1, y2, y3 in the above formula, we get:

y = (\[\frac{2+4+6}{3}\]) 

y = \[\frac{12}{3}\]

y = 4.

Therefore, the required coordinates of the centroid of triangle ABC is G (3, 4).

2. In an Equilateral Triangle ABC, G is the Centroid. What is the relationship between the Areas of ∆ GAB, ∆ GBC and ∆ GAC?

Solution: ar(∆ GAB) = ar(∆ GBC) = ar(∆ GAC)

3. What will be the position of the centroid in an isosceles right-angled triangle?

(a) On the hypotenuse of the triangle

(b) Inside

(c) Outside

(d) None of these

Solution: (b)

The centroid lies inside an isosceles right-angled triangle.

4. In a Triangle, the Centroid divides Medians of the Triangle in the Ratio.

(a) 1:2

(b) 2:3

(c) 2:1

(d) 1:3

Solution: (c)

The centroid of a triangle divides the medians of the triangle in the ratio 2:1.

5. In an Equilateral Triangle, the Lengths of Three Medians will be,

(a) Different

(b) can’t say

(c) same

(d) none of these

Solution: (c)

In an equilateral triangle, the lengths of the three medians are the same.