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Centroid: Definition, Formula, Properties, Practice Problems and FAQs

In a triangle, there are many centres, unlike a circle which has only one centre. Some of the most common centres of a triangle are centroid, incentre,ex-centre, circumcentre and orthocentre. In this article, we are going to learn about the centroid of a triangle in detail. The centroid, also known as the geometrical centre is basically the point of intersection of the medians i.e the line joining the mid-point of sides and the opposite vertices.

Table of Contents

  • Definition of centroid
  • Properties of Centroid
  • Centroid Formula
  • Practice Problems
  • FAQs

Definition of centroid

The centroid of a triangle is defined as a point of concurrency of the medians i.e, lines joining the mid-point of sides and the opposite vertices of a triangle. It is generally denoted by G. Referring to the figure above, ABC is a triangle AD, BE and CF are the medians of the triangle which intersect at a common point ‘G’ known as the centroid of the triangle ABC . Here points D, E, and F are the mid-points of the side BC, CA and AB respectively.

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Properties of Centroid

Following are the important properties of the centroid of a triangle -

  • It is located at the point of intersection of the medians
  • Centroid always divides the median internally in the ratio of 2:1 from the vertex.
  • Centroid divides the triangle into three parts of equal areas
  • The centroid of a triangle always lies inside the triangle

Centroid Formula

Given the coordinates of a triangle as A(x1,y1), B(x2,y2)and C(x3,y3), then the coordinates of the centroid is given by-

{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mi>G</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub></mrow><mn>3</mn></mfrac><mo>,</mo><mfrac><mrow><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><msub><mi>y</mi><mn>2</mn></msub><mo>+</mo><msub><mi>y</mi><mn>3</mn></msub></mrow><mn>3</mn></mfrac></mrow></mfenced></mstyle></math>","truncated":false}

=i=13xi3,i=13yi3

Derivation for finding the centroid of a triangle (Proof)

Let ABC be a triangle having coordinates A(x1,y1), B(x2,y2) and C(x3,y3) and G(x,y) be the centroid of the triangle.

Since, AD, BE and CF are the medians of the triangle then D, E and F become the mid-point of the side BC, AC and AB respectively.

As D is the mid-point of BC then using mid-point formula co-ordinates of D is given by -

{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mi>D</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub></mrow><mn>2</mn></mfrac><mo>,</mo><mfrac><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>+</mo><msub><mi>y</mi><mn>3</mn></msub></mrow><mn>2</mn></mfrac></mrow></mfenced></mstyle></math>","truncated":false}

From the properties of centroid, we know that G divides the median in the ratio of 2:1

Therefore, on applying section formula on the median AD we have,

{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mfenced><mfrac><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub></mrow><mn>2</mn></mfrac></mfenced><mo>+</mo><mn>1</mn><mfenced><msub><mi>x</mi><mn>1</mn></msub></mfenced></mrow><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></mfrac><mspace linebreak=\"newline\"/><mi>x</mi><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub></mrow><mn>3</mn></mfrac></mstyle></math>","truncated":false}{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><mi>y</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfrac><mrow><mn>2</mn><mfenced><mfrac><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>+</mo><msub><mi>y</mi><mn>3</mn></msub></mrow><mn>2</mn></mfrac></mfenced><mo>+</mo><mn>1</mn><mfenced><msub><mi>y</mi><mn>1</mn></msub></mfenced></mrow><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></mfrac><mspace linebreak=\"newline\"/><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mi>y</mi><mo>=</mo><mo>&#xA0;</mo><mfrac><mrow><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><msub><mi>y</mi><mn>2</mn></msub><mo>+</mo><msub><mi>y</mi><mn>3</mn></msub></mrow><mn>3</mn></mfrac></mstyle></math>","truncated":false}

Therefore, co-ordinates of the centroid (G) is given by   {"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mi>G</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub></mrow><mn>3</mn></mfrac><mo>,</mo><mfrac><mrow><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><msub><mi>y</mi><mn>2</mn></msub><mo>+</mo><msub><mi>y</mi><mn>3</mn></msub></mrow><mn>3</mn></mfrac></mrow></mfenced></mstyle></math>","truncated":false}

Practice Problems

  1. If are the roots of the equation then, find the centroid of Δ ABC  having coordinates as A(x 1, 1/ x 1), B(x2 ,1/ x2 ),C( x3 , 1/ x3 ) as shown in the figure given below

Solution-   Given,  are the roots of the equation then ,

Sum of roots
 Sum of product of roots taken two at a time  

Product of roots

We know that, centroid of a triangle G(x,y) is given by  




Therefore coordinates of centroid are 

  1. In a ABC, coordinates of B = (0,0) , AB= 2 units , ∠ABC = 600 and mid-point of BC = (2,0) , then find the centroid of ABC.
    Solution - Let D be the mid-point of BC

∴ coordinates of D=(2,0)

Hence, coordinates of C(x3,0) becomes (4,0)[∵BD = DC]

Now, in Δ ABD, we have
AB=BD= 2 units

∴ ∠ADB = ∠BAD = θ (say) [Angles opposite to equal sides are equal]

Also, in Δ ABD

60+ = θ  + θ = 1800     [Angle sum property of a triangle]

θ = 600

Therefore, Δ ABD becomes an equilateral triangle

So, AD = 2 units

Now, applying distance formula between A and B, we have

Squaring both sides

Again applying distance formula between A and D

Squaring both sides

Solving equation (1) and (2)



___________________



Putting = 1 in equation (1)


Therefore, coordinates of )

Note- ) is neglected because in the problem ABC is given as 600 so , the coordinate of A is going to lie in the first quadrant.

Now, the centroid of  Δ ABC i.e, G(x,y) =

=

=

  1. Find the centroid of the triangle made by the lines x + y = 2 and x y = 0 .

Solution: Given
x . y = 0
∴ either x = 0 or y = 0

Therefore, the lines are x = 0 , y = 0 and x + y = 2

The centroid of the triangle G(x,y) =

                                                              =

                                                               =

  1. If (1, 1) are the co-ordinates of the centroid of a triangle whose vertices are (0, 4), (-2,0) and (p, q). Determine the value of p,q

Solution:

Given (x1,y1)=(0,4)
(x2,y2) = (-2,0)
(x3,y3) = (p,q)

We know that, Centroid of a triangle is given by G(x,y) =

Hence,             and

              5                                q= -1 

FAQs

Ques 1) If ‘P is any internal point of a triangle such that area of △APB, △BPC and △APC are equal then P must be a ………………..
Answer:
As ‘P’ is an internal point of the triangle and it is dividing the triangle into three parts of equal area then P must be the centroid of the triangle.

Ques 2) What will be the location of the centroid in a right isosceles triangle? (inside/outside/on the hypotenuse/ at right-angled vertex)
Answer:
The position of a centroid is always inside the triangle as it is made by the intersection point of medians.

Ques 3) What is the relation between orthocentre(H), centroid(G) and circumcentre(O)?
Answer:
Orthocentre, centroid and circumcentre are always collinear and the centroid divides the line joining it in the ratio of 2:1 internally (except in an equilateral triangle)

Ques 4) What is the relation between orthocentre, centroid and circumcentre in the case of an equilateral triangle?
Answer:
In the case of an equilateral triangle, all three points centroid, circumcentre and orthocentre lie on the same point or coincide with each other.

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