# Centroid of a Triangle: Formula, Derivation, Properties, Example (2023)

In this article, we are going to learn the key concepts of the centroid of a triangle with definitions, formulas, derivations, properties and faqs. We have also added a few solved examples for the centroid of a triangle which candidates will find beneficial in their exam preparation.

Centroid meaning in hindi is केन्द्रक (kendrak). The most significant feature of a triangle is that the sum of the internal angles of a triangle is equivalent to 180 degrees. This is known as the angle sum property of a triangle. Consider a triangle with 3 vertices says P, Q, and R are represented as △PQR (where △ represent the symbol for triangle).

## What is the Centroid of a Triangle?

Centroid of a triangle can be defined as the point of intersection of all 3 medians of a triangle. The centroid of a triangle distributes all the medians in a 2:1 ratio. In other words, it is the point of intersection of all 3 medians.

Median is defined as a line that connects the midpoint of a side and the opposite vertex of the triangle.

The median is divided in the ratio of 2: 1 by the centroid of the triangle. It can be obtained by taking the average of x- coordinate locations and y-coordinate points of all the vertices of the triangle.

Triangles can be classified either on the basis of their angle or on the basis of the length of their sides. Below is the image for various types of triangles based on the classification.

## Centroid of a Triangle Formula

The formula for the centroid of a triangle is used to find the coordinates of the centroid of a triangle, for which the coordinates of vertices of the triangle are known.

Let coordinates of the vertices of a triangle are $$(x_{1},y_{1})$$, $$(x_{2},y_{2})$$, and $$(x_{3},y_{3})$$. Then the formula of the centroid of a triangle is given by

$$\textrm{Centroid}=\left[\frac{x_{1}+x_{2}+x_{3}}{3},\frac{y_{1}+y_{2}+y_{3}}{3}\right]$$.

Observe the below figure which shows the vertices of the triangle in the form of coordinates.

## How to Find the Centroid of a Triangle?

The centroid of a triangle formula is applied to find the centroid of a triangle using the coordinates of the vertices of a triangle. The formula for the centroid of the triangle is as follows:

$$Centroid=C\ (x,y)=\ \frac{\left(x_1+x_2+x_3\right)}{3},\ \frac{\left(y_1+y_2+y_3\right)}{3}$$

$$Here,\ x_1,\ x_2\ and\ x_3\ are\ the\ x-coordinates\ of\ the\ vertices\ of\ the\ triangle.$$

$$and\ y_1,\ y_2\ and\ y_3\ are\ the\ y-coordinates\ of\ the\ vertices\ of\ the\ triangle.$$

The coordinates of the centroid of a triangle can simply be determined if we know the coordinates of the vertices of the triangle. By applying the section formula the centroid of a triangle can be derived as follows.

$$Consider\ ABC\ to\ be\ a\ triangle\ with\ the\ coordinates\ of\ vertices\ as:$$

$$A(x_1,y_1),\ B(x_2,\ y_2),\ and\ C(x_3,y_{3\ }),\$$

$$such\ that\ D\ ,\ E,\ and\ F\ are\ midpoints\ of\ the\ side\ AB,\ BC,and\ AC\ respectively.$$

$$The\ centroid\ of\ a\ triangle\ is\ denoted\ by\ G.$$

$$As,\ D\ is\ the\ midpoint\ of\ side\ AB,\ u\sin g\ the\ midpoint\ formula,\$$

$$we\ get\ its\ coordinates\ as,$$

$$D=\ \frac{\left(x_1+x_2\right)}{2}$$

$$As\ studied\ the\ centroid\ of\ a\ triangle\ divides\ the\ medians\ in\ the\ ratio\ 2:1.$$

$$Therefore,\ from\ the\ coordinates\ of\ D,we\ can\ find\ the\ coordinates\ of\ G\ as,$$

$$X-coordinateofG:$$

$$\Rightarrow\frac{\left[\frac{2\left(x_1+x_2\right)}{2}+1\left(x_3\right)\right]}{2+1}=\frac{\left(x_1+x_2+x_3\right)}{3}$$

$$Y-coordinate\ of\ G:$$

$$\Rightarrow\frac{\left[\frac{2\left(y_1+y_2\right)}{2}+1\left(y_3\right)\right]}{2+1}=\frac{\left(y_1+y_2+y_3\right)}{3}$$

$$Hence,\ the\ coordinates\ of\ G\ are\ given\ as,$$

$$\frac{\left(x_1+x_2+x_3\right)}{3},\ \frac{\left(y_1+y_2+y_3\right)}{3}$$

The centroid of a right-angle triangle is the point of intersection of three medians, induced from the vertices of the triangle to the midpoint of the opposite sides. The centroid of an equilateral triangle; in an equilateral triangle the orthocenter, circumcenter of a triangle, centroid and incenter of a triangle coincide. This states that the center of the circle is the centroid and height coincides with the median.

Question: Find the centroid of a triangle whose vertices are (6,3), (4,5), and (5,4).

Solution: To find the Centroid of a triangle.

$$Given\ parameters\ are,\$$

$$\Rightarrow(x_1,\ y_1)=(6,3)\$$

$$\Rightarrow\ (x_2,\ y_2)=(4,5)$$

$$\Rightarrow\ (x_3,\ y_3)=(5,4)$$

$$U\sin g\ centroid\ formula,$$

$$The\ centroid\ of\ a\ triangle=\left[\frac{\left(x_1+x_2+x_3\right)}{3},\ \frac{\left(y_1+y_2+y_3\right)}{3}\right]$$

$$\Rightarrow\ \left[\frac{\left(6+4+5\right)}{3},\frac{\left(3+5+4\right)}{3}\right]$$

$$\Rightarrow\frac{15}{3},\frac{12}{3}$$

$$The\ centroid\ of\ a\ triangle\ for\ the\ given\ vertices\ are=(5,4)$$

Question: Find the centroid of a right-angled triangle whose vertices are (0,6), (6,0), and (0,0).

Solution: To find the centroid of a right-angled triangle.

$$Given\ parameters\ are,\$$

$$\Rightarrow(x_1,\ y_1)=(0,6)\$$

$$\Rightarrow\ (x_2,\ y_2)=(6,0)$$

$$\Rightarrow\ (x_3,\ y_3)=(0,0)$$

$$U\sin g\ centroid\ formula,$$

$$The\ centroid\ of\ a\ triangle=\left[\frac{\left(x_1+x_2+x_3\right)}{3},\ \frac{\left(y_1+y_2+y_3\right)}{3}\right]$$

$$\Rightarrow\ \left[\frac{\left(0+6+0\right)}{3},\frac{\left(6+0+0\right)}{3}\right]$$

$$\Rightarrow\frac{6}{3},\frac{6}{3}=\left(2,2\right)$$

$$The\ centroid\ of\ a\ right-angled\ triangle\ for\ the\ given\ vertices\ are=(2,2)$$

## Derivation for Centroid of a Triangle’s Formula

Consider a triangle ABC, whose coordinates are A$$(x_{1},y_{1})$$, B$$(x_{2},y_{2})$$, and C$$(x_{3},y_{3})$$. $$D$$, $$E$$, and $$F$$ are the midpoints of the sides $$AB$$, $$BC$$, and $$CA$$ respectively. The centroid of a triangle is represented as ‘$$G$$’.

As $$D$$ is the midpoint of the side $$BC$$, by using the midpoint formula the coordinates of $$D$$ can be determined as:

$$\left[\frac{x_{2}+x_{3}}{2},\frac{y_{2}+y_{3}}{2}\right]$$.

Since we know that the point‘$$G$$ divides the median $$AD$$ in the ratio of $$2:1$$. Therefore, the coordinates of the centroid $$G$$’ can be determined by using the section formula.

X-coordinates of $$G$$:

By using the section formula, the $$x$$-coordinates of ‘$$G$$’ are given as:

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

$$\Rightarrow$$ $$x = \frac{[x_{2}+x_{3}+x_{1}]}{3}$$

$$\Rightarrow$$ $$x = \frac{[x_{1}+x_{2}+x_{3}]}{3}$$

Y-coordinates of $$G$$:

Similarly, by using the section formula, the $$y$$-coordinates of ‘$$G$$’ are given as:

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

$$\Rightarrow$$ $$y = \frac{[y_{2}+y_{3}+y_{1}]}{3}$$

$$\Rightarrow$$ $$y = \frac{[y_{1}+y_{2}+y_{3}]}{3}$$

Therefore, the coordinates of the centroid ‘$$G$$’ is $$\left[\frac{x_{1}+x_{2}+x_{3}}{3},\frac{y_{1}+y_{2}+y_{3}}{3}\right]$$.

## Properties of Centroid of a Triangle

Some of the useful properties of the centroid of a triangle are as follows:

• The centroid is also recognized as the geometric center of the structure.
• Centroid of a triangle is formed by the intersection of the medians of the triangle.
• The centroid internally distributes all three medians in a ratio 2:1.
• The centroid of a triangle always lies within the triangle.

• The Centroid theorem says that the centroid of a triangle is at a 2/3 length from the vertex of a triangle and at a measure of 1/3 from the side opposite to the vertex.

Now, you have learned the centroid of a triangle, try to attempt Centroid MCQs.

## Relation Between Orthocentre, Centroid and Circumcentre

• The orthocenter is the location where the three altitudes of a triangle meet. A line segment developed from one vertex to the opposite side, which is perpendicular to the opposite side, is known as the altitude of a triangle.
• The centroid as known is the point of intersection of the three medians. A median is where each of the straight lines joins the midpoint of a side with the opposite vertex.
• The circumcenter is the point of junction of the three perpendicular bisectors. The perpendicular bisector of a triangle is the lines drawn perpendicularly from the midpoint of the triangle.
• The Centroid of a triangle divides the line joining circumcentre and orthocentre in the ratio 1:2.

Consider H, O and G to be the orthocentre, circumcentre and centroid of any triangle.

Here, G divides the line segment OH beginning from O in the ratio of 1:2

internally,

i.e., OG/GH=1:2

## Difference Between a Median and Centroid of a Triangle

The differences between median and centroid of a triangle are tabulated below:

 Median of a Triangle Centroid of a Triangle A median of a triangle is a line segment that joins a vertex to the midpoint of the side that is opposite to that vertex. The centroid of a triangle is the point of intersection of all the three medians of a triangle. A median divides the triangle into two triangles which are of equal area. The centroid of a triangle is always within a triangle. 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 $$2:1$$.

## Difference Between Orthocentre and Centroid of a Triangle

The orthocenter is the junction point of the altitudes whereas the centroid is the intersection position of the medians. An orthocenter may lie outside of the triangle but a centroid always lies inside the triangle.

## Difference Between Incentre and Centroid of a Triangle

As we know the centroid is the intersection position of the median, however, the incenter is the intersection point of the angle bisectors. Both the centroid and incenter lie inside the triangle.

We hope that the above article on Centroid of a Triangle is helpful for your understanding and exam preparations. Stay tuned to the Testbook app for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.

 If you are checking Centroid of a Triangle article, also check the related maths articles in the table below: Area of a Triangle Triangular Numbers Polygons Equilateral Triangle Mensuration 2D Triangles

## Centroid of a Triangle FAQs

Q.1What is the centroid of a triangle?

Ans.1 The centroid of a triangle can be defined as the point of intersection of all the three medians of a triangle. The centroid of a triangle distributes all the medians in a 2:1 ratio.

Q.2Is a centroid equidistant from vertices?

Ans.2 The centroid intersects the vertices at the middle of the triangle, therefore it is the point that is equal in distance from all three vertices.

Q.3What is the relation between orthocentre centroid and circumcentre?

Ans.3 The Centroid of a triangle divides the line joining circumcentre and orthocentre in the ratio 1:2. Consider H, O and G to be the orthocentre, circumcentre and centroid of any triangle. Here, G divides the line segment OH beginning from O in the ratio of 1:2 internally, i.e., OG/GH=1:2

Q.4How do you find the centroid of a triangle with 3 points?

Ans.4 Step 1: Recognize the coordinates of every vertex. Step 2: Combine all the x values from the three vertices coordinates and divide by three. Step 3: Calculate the total of all the y values from the 3 vertices coordinates and divide by three. Step 4: The values obtained in the second and third steps give the centroid coordinate.

Q.5What is the difference between orthocenter, circumcenter and centroid?

Ans.5 The centroid lies in between the orthocenter and the circumcenter. Secondly, the interval between the centroid and the orthocenter is always twice the length between the centroid and the circumcenter. An orthocenter may lie outside of the triangle but a centroid always lies inside the triangle.

Q.6Does centroid divides the median in the equal ratio?

Ans.6 Centroid divides each median into two parts, which are always in the ratio 2:1.

Q.7What is the difference between centroid and centre of gravity?

Ans.7 Centre of gravity of triangle of any object is the point where gravity acts on the body while centroid is referred to as the geometrical centre of a uniform density object.

Q.8How to find length of centroid of equilateral triangle?

Ans.8 Centre of mass or centroid of equilateral triangle is at a distance of H/3 from the centre of the base of the triangle.

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