Centroid of a Triangle: A centroid is the centre of gravity present inside an object. In the case of a triangle, the centroid is the centre point of the triangle.

The centroid of a triangle is the point of intersection of its three medians. In coordinate geometry, it is calculated by taking \(x\) and \(y\) coordinates of vertices of a triangle. Read on to find out more about the Centroid of a Triangle!

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Centroid of a Triangle

The centroid of a triangle is the point of intersection of the medians of a triangle. Medians are the line segments, which are drawn from the vertex to the mid-point of the opposite side of the vertex of a triangle. Each median of the triangles divides the triangle into two small triangles of equal area.

The point of concurrency of all medians of the triangle is known as the centroid of the triangle. The centroid of a triangle always lies inside it, unlike other point concurrencies such as orthocentre, circumcentre, etc.

Definition of Centroid of a Triangle

The centroid of a triangle is the centre point of the triangle. The centroid of the triangle is a point formed by the intersection of the medians of a triangle. Centroid is also known as the centre of gravity of the polygon. Centroid is the most commonly known point of concurrency.

The above figure shows the centroid of a triangle by constructing the medians. Medians are drawn by joining the line segment from the vertex of the triangle to the mid-point of the opposite side of the triangle.
Centroid divides the median in the ratio \(2 : 1\).

Properties of Centroid of a Triangle

Some of the properties of the centroid of a triangle are listed below. These properties of the centroid of a triangle help to distinguish the centroid from all other points of concurrence.

1. The centroid of a triangle is also known as the geometrical centre of the triangle.
2. The centroid of the triangle is the point of intersection of all three medians of the triangle.
3. The centroid of a triangle is always located inside the triangle.
4. Centroid, circumcentre, incentre, and orthocentre are the four different points of concurrency based on different criteria in a triangle. Among those, the centroid is the most widely used point of concurrency.
5. Centroid divides the median in the ratio \(2 : 1\).

Centroid of a Triangle Formula

This formula 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 centroid of a triangle is given by
Centroid \( = \left[{\frac{{{x_1} + {x_2} + {x_3}}}{3},\frac{{{y_1} + {y_2} + {y_3}}}{3}} \right]\)
The below figure gives the centroid formula whose coordinates of the vertices are given:

Derivation of Centroid of a Triangle Formula

Consider a triangle \(ABC\), whose coordinates are \(A(x_1, y_1)\), \(B(x_2, y_2)\) and \(C(x_3, y_3)\). \(D, E, F\) are the midpoints of the sides \(AB, BC, CA\).

By using the midpoint formula, the coordinates of the point \(D\) are calculated as \(\left[{\frac{{{x_2} + {x_3}}}{2},\frac{{{y_2} + {y_3}}}{2}} \right]\)
From the property of centroid \((G)\), it divides median \(AD\) in the ratio \(2 : 1\).

By using the section formula, the coordinates of the centroid are given below:
\({G} = \left\{ {\frac{{\left[ {2\left( {\frac{{{x_2} + ~{x_3}}}{2}~} \right) + 1\left( {{x_1}} \right)} \right]}}{{2 + 1}}~,~\frac{{\left[ {2\left( {\frac{{{y_2} + ~{y_3}}}{2}~} \right) + 1\left( {{y_1}} \right)} \right]}}{{2 + 1}}} \right\}\)

Centroid Theorem

Let \(PQR\) is a triangle having centroid \(V\) and \(S, T, U\) are the sides of the triangle. Centroid Theorem of a triangle states that the centroid of the triangle is at \(\frac{2}{3} \rm{rd}\) of the distance from the vertex to the mid-point of the opposite side.

By using the theorem:
\( {{QV}} = \frac{2}{3} {{QU}}\); \( {{PV}} = \frac{2}{3} {{PT}}\); \({{RV}} = \frac{2}{3}{{RS}}\)

Difference Between Centroid and Incentre of a Triangle

Centroid and Incentre of the triangle are the points of concurrency of the triangles, which has differences between them depending upon the triangle it lies. Some of the main differences between Centroid and Incentre are listed below:

Centroid	Incentre
The centroid of a triangle is the point of intersection of medians of the triangle.	The incentre of the triangle is the point of intersection of the angle bisectors of the triangle.
The centroid of a triangle always lies inside the triangle.	The incentre of a triangle always lies inside the triangle.
The centroid of a triangle divides the median in the ratio \(2 : 1\)	There is no fixed ratio, into which it divides the angle bisectors

Difference Between Centroid and Orthocentre of a Triangle

Centroid and Orthocentre of the triangle are the points of concurrency of the triangles, which has differences between them depending upon the triangle it lies. Some of the main differences between Centroid and Orthocentre are listed below:

Centroid	Orthocentre
The centroid of a triangle is the point of intersection of medians of the triangle.	The orthocentre of the triangle is the point of intersection of the altitudes of the triangle.
The centroid of a triangle always lies inside the triangle.	The orthocentre of a triangle may lie both inside and outside the triangle.
The centroid of a triangle divides the median in the ratio \(2 : 1\)	There is no fixed ratio into which it divides the altitudes.

Difference Between Circumcentre and Centroid of a Triangle

Centroid and Circumcentre of the triangle are the points of concurrency of the triangles, which has differences between them depending upon the triangle it lies. Some of the main differences between Centroid and Circumcentre are listed below:

Centroid	Circumcentre
The centroid of a triangle is the point of intersection of medians of the triangle.	The circumcentre of the triangle is the point of intersection of the perpendicular bisectors of the sides of the triangle.
The centroid of a triangle always lies inside the triangle.	The circumcentre of a triangle may lie both inside and outside the triangle.
The centroid of a triangle divides the median in the ratio \(2 : 1\)	Circumcentre does not divide the perpendicular bisector at any particular ratio.

Centroid of a Triangle: Solved Examples

Q.1. Find the centroid of the triangle whose vertices are \(A(4, 8)\), \(B(3, 9)\), and \(C(5, 10)\).
Ans: Given vertices of the triangle are \(A(4, 8)\), \(B(3, 9)\), \(C(5, 10)\).
The formula for calculating the coordinates of the centroid of a triangle is
\({G} = \left[{\frac{{{x_1} + {x_2} + {x_3}}}{3},\frac{{{y_1} + {y_2} + {y_3}}}{3}} \right]\)
\( \Rightarrow {{G}} = \left[{\frac{{4 + 3 + 5}}{3},\frac{{8 + 9 + 10}}{3}} \right]\)
\( \Rightarrow {{G}} = \left({\frac{{12}}{3},\frac{{27}}{3}} \right)\)
\( \Rightarrow G = (4, 9)\)
Hence, the coordinate of the centroid is \((4, 9)\).

Q.2. Determine the centroid of a right-angled triangle using the centroid formula, if the vertices of the triangle are \((0, 5)\), \((5,0)\), and \((0, 0)\).
Ans: Given vertices of the triangle are \((0, 5)\), \((5, 0)\), and \((0, 0)\).
The formula for calculating the coordinates of the centroid of a triangle is
\({{G}} = \left[{\frac{{{x_1} + {x_2} + {x_3}}}{3},\frac{{{y_1} + y + {y_3}}}{3}} \right]\)
\( \Rightarrow { {G}} = \left[{\frac{{0 + 5 + 0}}{3},\frac{{5 + 0 + 0}}{3}} \right]\)
\( \Rightarrow {{G}} = \left({\frac{5}{3},\frac{5}{3}} \right)\)
Hence, the coordinate of the centroid is \(\left({\frac{5}{3},\frac{5}{3}} \right)\)

Q.3. \(G\) is the centroid of the equilateral triangle \(ABC\). If \(AB = 10\,\rm{cm}\), find the length of \(AG\).
Ans: Given: \(G\) is the centroid of the equilateral triangle.
Given, \(AB = 10\,\rm{cm}\)
So, \({{AB}} = {{BC}} = {{AC}} = 10\,{\text{cm}}\)
Here, \(AD\) is the median, \(D\) is the mid-point of side \(BC\).
\(BD = CD = \frac{{BC}}{2} = 5\,{\text{cm}}\)

In an equilateral triangle, the medians are perpendicular to the respective sides. So, \(AD ⊥ BC\).
By using Pythagoras Theorem:
In triangle \(ADB\),
\(A {D^2} = A {B^2} – B {D^2}\)
\( \Rightarrow AD = \sqrt { { {10}^2} – {5^2}} = 5\sqrt 3 \, {\text{cm}}\)
We know that centroid G divides median AD in the ratio \(2 : 1\).
\(AG = \frac{2}{3}AD = \frac{ {10\sqrt 3 }}{3}\, {\text{cm}}\)
Hence, the value of \(AD\) is \(\frac{ {10\sqrt 3 }}{3}\, {\text{cm}}\)

Q.4. The coordinates of the vertices of a triangle are \((4, – 3)\), \((- 5, 2)\) and \((x, y)\). If the centre of gravity of the triangle is at the origin, find \(x, y\).
Ans: Given centroid of the triangle is at the origin.
Thus, \(G = (0, 0)\)
Given coordinates of the vertices of a triangle are \((4, – 3)\), \((- 5, 2)\) and \((x, y)\).
Use the formula for calculating the coordinates of the centroid of a triangle is
\(G = \left[{\frac{{{x_1} + {x_2} + {x_3}}}{3},\frac{{{y_1} + y + {y_3}}}{3}} \right]\)
\( \Rightarrow (0,0) = \left[{\frac{{4 – 5 + x}}{3},\frac{{ – 3 + 2 + y}}{3}} \right]\)
\( \Rightarrow (0,0) = \left[{\frac{{x – 1}}{3},\frac{{{\text{y}} – 1}}{3}} \right]\)
Equating the coordinates of both points
\( \Rightarrow \frac{{x – 1}}{3} = 0 \) and \(\frac{{y – 1}}{3} = 0\)
\(\Rightarrow x – 1 = 0\) and \(y – 1 = 0\)
\(\Rightarrow x = 1\) and \(y = 1\)

Q.5. The centroid of a triangle is \((- 1, – 2)\), and coordinates of its two vertices are \((4, 6)\) and \((- 8, – 12)\). Find the coordinates of its third vertex.
Ans: Let the coordinates of the third vertex is \((x, y)\)
Thus, \(G = (- 1, – 2)\)
Given coordinates of the vertices of a triangle are \((4, 6)\), \((- 8, – 12)\).
Use the formula for calculating the coordinates of the centroid of a triangle is
\(G = \left[{\frac{{{x_1} + {x_2} + {x_3}}}{3},\frac{{{y_1} + y + {y_3}}}{3}} \right]\)
\( \Rightarrow ( – 1, – 2) = \left[{\frac{{4 – 8 + x}}{3},\frac{{6 – 12 + y}}{3}} \right]\)
\( \Rightarrow ( – 1, – 2) = \left[{\frac{{x – 4}}{3},\frac{{y – 6}}{3}} \right]\)
Equating the coordinates of both points
\( \Rightarrow \frac{{x – 4}}{3} = – 1\) and \(\frac{{y – 6}}{3} = – 2\)
\( \Rightarrow x – 4 = – 3\) and \(y – 6 = – 6\)
\( \Rightarrow x = 4 – 3\) and \(y = – 6 + 6\)
\( \Rightarrow x = 1\) and \(y = 0\)
Hence, the coordinates of the third vertex of the triangle are \((1, 0)\).

Summary

In this article, we have discussed that Centroid is the geometric centre of a two-dimensional plane surface. Centroid is the point of intersection of medians of the triangle. We have also discussed the difference of centroid with other important points in the triangle. This is one of the most important concepts in a triangle and in geometry, as a triangle is a basic shape in geometry. In this article, we have provided detailed information on Centroid.

FAQs on Centroid of a Triangle

Check most commonly asked questions about the centroid of a triangle below:

Q.1. What is meant by ‘the centroid of a triangle at \(\frac{1}{3}\)’?
Ans: According to the centroid theorem, the Centroid is present at \(\frac{1}{3} \rm{rd}\) of the distance from the mid-point of the opposite side of the vertex.

Q.2. What is the centroid of a triangle?
Ans: The Centroid is a point inside a triangle where all the medians intersect each other.

Q.3. Where does the centroid of a triangle lie?
Ans: The Centroid of the triangle lies inside the triangle.

Q.4. What is the formula of Centroid?
Ans: The formula to find the Centroid of a triangle is given below:
\(G = \left[{\frac{{{x_1} + {x_2} + {x_3}}}{3},\frac{{{y_1} + y + {y_3}}}{3}} \right]\)

Q.5. What is meant by ‘the centroid of a triangle at \(\frac{2}{3}\)’?
Ans: According to the centroid theorem, the Centroid is present at \(\frac{2}{3} \rm{rd}\) of the distance from the vertex.

Now you are provided with all the necessary information regarding the centroid of a triangle. Practice more questions and master this concept. Students can make use of NCERT Solutions for Maths provided by Embibe for their exam preparation.