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What is Centroid of a Triangle and the Properties?

Centroid of a Triangle

The centroid of a triangle can be defined as a point of concurrency of the medians of a triangle.

Medians can be defined as the line segments which are drawn from the vertex to the mid-point of the opposite side of the vertex. The median of a triangle divides the triangle into two smaller triangles which have equal areas. The point at which the medians of a triangle intersect is known as centroid. The centroid always occurs inside a triangle, unlike other points of concurrencies of a triangle.

What is Centroid of a Triangle?

The centroid of a triangle is the point at which three medians of a triangle intersect which is one of the four points of concurrencies of a triangle. The medians of a triangle are formed when the vertices of a triangle are joined with the midpoint of the opposite sides of the triangle. The centroid of the triangle divides the median in the ratio of 2:1. It can be calculated by taking the average of x-coordinate points and y-coordinate points of all the vertices of the triangle.

Properties of the Centroid of Triangle

• The centroid is also called the geometric center of the object.
• The centroid of a triangle is the point where all the three medians intersect with each other.
• The centroid of a triangle divides the median in the ratio 2:1.
• The centroid of a triangle lies within a triangle.
• It is the center of gravity.
• It is the center of the object.

Centroid of Triangle Formula

The centroid of a triangle formula is used for finding the centroid of a triangle by using the coordinates of the vertices of a triangle. The coordinates of the centroid of a triangle can be found out only when the coordinates of the vertices of the triangle are known. The formula for the centroid of the triangle is given below:

Where, are the ‘x-coordinates’ of the vertices of the triangle; and are the ‘y-coordinates’ of the vertices of the triangle.

Centroid Theorem

The centroid theorem says that the centroid of the triangle is at 2/3 of the distance from the vertex to the mid-point of the sides.

In the above figure, PQR is a triangle which have a centroid V where S, T and U are the midpoints of the sides of the triangle PQ, QR and PR, respectively. Therefore, as per the theorem:

QV = 2/3 QU, PV = 2/3 PT and RV = 2/3 RS

Centroid of a Right-Angle Triangle

The centroid of a right-angle trianglecan be defined as the point of intersection of three medians which comes from the vertices of the triangle to the midpoint of the opposite sides.

Centroid of a Square

The point where the diagonals of the square meet is the centroid of the square. We all know that square has all the four sides as equal. So, it is easy to locate the centroid in it. In the figure given below, 0 is the centroid of the square.

Centroid Formula

Let us consider a triangle where the three vertices of the triangle are . The centroid of a triangle can be found out by taking the average of X and Y coordinate points of all three vertices. Thus, the centroid of a triangle can be written as:

Examples on calculating Centroid

Below is given a solved example to calculate the centroid of triangles with the given values of vertices.

Question 1: Find out the centroid of the triangle whose vertices are A(2,6), B(4,9), and C(6,15).

Solution:

Given:

The formula to find out the centroid of a triangle is

Now, the given values are substituted in the formula.

Centroid of a triangle + ((2+4+6)/3, (6+9+15)/3) = (12/3, 30/3) = (4, 10)

Hence, the centroid of the triangle for the given vertices A(2,6), B(4,9), and C(6, 15) is (4, 10).

Now coming to the main part of this topic i.e., the centroid divides the median into two parts. If the larger part is 8cm, then what is the measure of the smaller part?

Now we all know that the centroid divides the median into two parts in the ratio 2:1.

Let us suppose that larger part = x

And smaller part = y

Therefore, x:y = 2:1

x/y = 2/1

x = 2y

Larger part = 2* smaller part

Smaller part = larger part/2

Therefore, in the above case,

x = 8cm(given)

Hence, smaller part, y = 8/2 = 4 cm.