In Mathematics, centroid defines the geometric center of a two-dimensional plane surface. It is a point that is located from the arithmetic mean position of all the points in the plane surface. Otherwise, it is defined as the average of all the points in the plane figure. The centroid can be found for different geometrical shapes. In this article, let us discuss in detail about the centroid of a triangle.
Centroid of a Triangle Definition
For a two-dimensional shape “triangle,” the centroid is obtained by the intersection of its medians. The line segments of medians join vertex to the midpoint of the opposite side. All three medians meet at a single point (concurrent). The point of concurrency is known as the centroid of a triangle.
From the given figure, three medians of a triangle meet at a centroid “G”. Centroid is also known as the centre of gravity.
Properties
The important properties of the centroid of a triangle are:
- The centroid of a triangle is located at the intersecting point of all three medians of a triangle
- It is considered one of the three points of concurrency in a triangle,
i.e., incenter, circumcenter, centroid - The centroid is positioned in the inside of a triangle
- At the point of intersection (centroid), each median in a triangle is divided in the ratio of 2: 1
Centroid of a Triangle Formula
The centroid triangle coordinates can be obtained using the given formula, If (x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3}) are the coordinates for the three vertices of a triangle, then
The centroid of a triangle = ((x_{1}+x_{2}+x_{3})/3 , (y_{1}+y_{2}+y_{3})/3 )
Where
x_{1}, x_{2}, x_{3} are the x coordinates of the vertices of a triangle.
y_{1}, y_{2}, y_{3} are the y coordinates of the vertices of a triangle.
Proof
Let ABC be a triangle with the vertex coordinates A( (x_{1}, y_{1}), B(x_{2}, y_{2}), and C(x_{3}, y_{3}). The midpoints of the side BC, AC and AB are D, E, and F, respectively. The centroid of a triangle is represented as “G.”
As D is the midpoint of the side BC, the midpoint formula can be determined as:
((x_{2}+x_{3})/2, (y_{2}+y_{3})/2)
We know that point G divides the median in the ratio of 2: 1. Therefore, the coordinates of the centroid “G” are calculated using the section formula
To find the x-coordinates of G:
X = (2(x_{2}+x_{3})/2 + 1.x_{1 })/ (2+1)
x= (x_{2}+x_{3}+x_{1})/3
x = (x_{1}+x_{2}+x_{3})/3
To find the y-coordinates of G:
Similarly, fo y-coordinates of the centroid “G.”
Y =(2(y_{2}+y_{3})/2 + 1.y_{1 })/ (2+1)
y= (y_{2}+y_{3}+y_{1})/3
x = (y_{1}+y_{2}+y_{3})/3
Therefore, the coordinates of the centroid “G” is ((x_{1}+x_{2}+x_{3})/3 , (y_{1}+y_{2}+y_{3})/3 )
Hence, proved
Example
Question:
Determine the coordinates of the centroid of a triangle whose vertices are (-1, -3), (2, 1) and (8, -4)
Solution:
Given: The vertices coordinates are (-1, -3), (2, 1) and (8, -4)
From this, we can write the x- coordinates
x_{1 }= -1, x_{2} = 2, x_{3 } = 8
Similarly for the y-coordinates
y_{1 }= -3, y_{2} = 1, y_{3 } = -4
The formula to find the centroid of a triangle is
G = ((x_{1}+x_{2}+x_{3})/3 , (y_{1}+y_{2}+y_{3})/3 )
Substitute the values, G = ((-1+2+8)/3 , (-3+1-4)/3)
G =( 9/3 , -6/3)
G = (3, -2)
Therefore, the centroid of a triangle, G = (3, -2)
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Related Links | |
Triangles | Average |
Median | Arithmetic Mean |