Properties of the Centroid of a Triangle

1. 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 triangle is at a 23distance from the vertex of a triangle and at 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 

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) = ((\[\frac{x1+x2+x3}{3}\]) , (\[\frac{y1+y2+y3}{3}\]))

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. (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 ((\[\frac{x2+x3}{3}\] , (\[\frac{y2+y3}{3}\]))

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 =  \[\frac{2(\frac{x2+x3}{2})+1(x1)}{2+1}\]

x = (\[\frac{x1+x2+x3}{3}\])

To Find the Y-Coordinates of G:

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

y = \[\frac{2(\frac{y2+y3}{2})+1(y1)}{2+1}\]

y = (\[\frac{y1+y2+y3}{3}\])

Therefore, the coordinates of the centroid G (x, y) is ((\[\frac{x1+x2+x3}{3}\]) , (\[\frac{x1+x2+x3}{3}\]))

Solved Examples:

Q.1. Find the coordinates of the centroid of a triangle ABC whose vertices are A (1, 2), B (3, 4) and C (5, 6).

Solution: 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:

x = (\[\frac{x1+x2+x3}{3}\]) 

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 = (\[\frac{y2+y3}{3}\])

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).

Q.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)

Q.3. What will be the position of the centroid in an isosceles right-angled triangle?
(a) on the hypotenuse of triangle
(b) inside
(c) outside
(d) none of these

Solution: (b)

The centroid lies inside an isosceles right-angled triangle.

Q.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.

Q.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 three medians are the same.