An Introduction to Centroid of a Triangle

The centroid of a triangle is an essential subject in geometry. A triangle is a three-sided, three-interior-angle delimited figure. A triangle can be categorised into numerous sorts based on its sides and angles.

• Scalene triangle

• Isosceles triangle

• Equilateral triangle

• Triangle with an acute angle

• Triangle with an obtuse angle

• Triangle with a right angle

A triangle’s centroid is an important attribute. Let’s look at the definition of centroid, as well as the formula, attributes, and centroid for various geometric shapes.

Definition of Centroid

The centroid is the object’s centre point. The centroid of any triangle is the point where the three medians of the triangle intersect. It’s also regarded as the intersection location of all three medians. The median is a line that links the midpoint of a side to the triangle’s opposite vertex. In a 2:1 ratio, the triangle’s centroid separates the median from the median. It can be calculated by averaging the x- and y-coordinate points of all the triangle’s vertices.

Theorem of Centroid

According to the centroid theorem, the triangle’s centroid is 2/3 of the distance between the vertex and the midpoint of the sides.

Centroid for Right angled Triangle

The centroid of a right angle triangle is the location where three medians, drawn from the triangle’s vertices to the opposite sides’ midpoints, coincide.

Centroid for Square

The centroid of the square is the place where the diagonals of the square cross. The square, as we all know, has equal sides on all sides. As a result, finding the centroid is simple

Characteristics of Centroid

The following are the properties of the centroid:

• The centroid is the object’s centre.
• It is the gravitational centre.
• It should always be contained within the object.
• It is the place where the medians agree.

The Centroid Formula

Consider the shape of a triangle. If the triangle’s three vertices are A(x₁ , y₁), B(x₂, y₂), and C(x₃, y₃), the centroid can be computed by averaging the X and Y coordinate points of all three vertices. As a result, the triangle’s centroid can be written as:

Triangle centroid = ((x₁ +x₂+x₃)/3, (y₁+y₂+y₃)/3)

Derivation of the Centroid Formula

Let ABC be a triangle with the vertices A((x₁ , y₁), B(x₂, y₂), and C(x₃, y₃) (x₃, y₃). D, E, and F are the midpoints of the sides BC, AB, and AC, respectively. The centroid of the triangle is denoted by the letter “G.”

Because D is the midpoint of side BC, the midpoint formula is as follows:

((x₂+x₃)/2, (y₂+y₃)/2)

We already know that point G splits the median in a 2:1 ratio. As a result, the section formula is used to obtain the coordinates of the centroid “G.”

G’s x-coordinates are as follows:

x = (2(x₂+x₃)/2 + 1.x₁ )/ (2+1)

x= (x₂+x₃+x₁ )/3

x = (x₁ +x₂+x₃)/3

G’s y-coordinates are as follows:

Similarly, for the centroid’s y-coordinates “G.”

y =(2(y₂+y₃)/2 + 1.y₁)/ (2+1)

y= (y₂+y₃+y₁)/3

y = (y₁+y₂+y₃)/3

As a result, the coordinates of the centroid “G” is ((x₁ +x₂+x₃)/3, (y₁+y₂+y₃)/3.

Examples based on Centroid of a Triangle

1. Determine the coordinates of the centroid of a triangle with vertices (-1, -3), (2, 1), and (3, 1). (8, -4)

Solution:

Given,

The coordinates of the vertices are (-1, -3), (2, 1), and (8, -4)

We can write the x-coordinates from this.

x₁ = -1, x₂ = 2, x₃ = 8

Likewise, with the y-coordinates;

y₁ = -3, y₂ = 1, y₃ = -4

The formula for finding the triangle’s centroid is

G = ((x₁ +x₂+x₃)/3 , (y₁+y₂+y₃)/3)

Replace the values with G = ((-1+2+8)/3, (-3+1-4)/3.

G =(9/3 , -6/3)

G = (3, -2)

As a result, the triangle’s centroid, G = (3, -2)

2. Find the centroid of a triangle with the vertices A(2, 6), B(4, 9) and C(4, 9). (6,15)

Solution:

Given:

A(x₁ , y₁) = A (2, 6)

B(x₂, y₂) = B (4,9)

C(x₃, y₃) = C (6,15)

The formula for finding the triangle’s centroid is = ((x₁ +x₂+x₃)/3, (y₁+y₂+y₃)/3)

Replace the supplied values in the formula now.

((2+4+6)/3, (6+9+15)/3) triangle centroid

= (12/3, 30/3)

= (4, 10)

As a result, the triangle’s centroid for the specified vertices A(2, 6), B(4,9), and C(6,15) is (4, 10).

3. PQR is (2, 1), (3, 2), and (-2, 4). Then find its centroid.

Solution:

The vertices of triangle pQR are (2, 1), (3, 2), and (-2, 4).

We know the formula for the centroid;

((x₁ +x₂+x₃)/3, (y₁+y₂+y₃)/3) centroid

We get when we combine the values;

O = (2+3-2)/3, (1+2+4)/3, centroid

O = (3/3, 7/3)

O = (1, 7/3)

As a result, (1, 7/3) is the centroid of the triangle with vertices (2, 1), (3, 2), and (-2, 4).

Conclusion

This article has provided enormous information regarding the centroid of a triangle. It has also included the various formulae attached to the concept of centroid of triangle. In addition to this, the article has also learnt the properties of the centroid of a triangle along with which few examples have been discussed.