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.
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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.
Following are the important properties of the centroid of a triangle -
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-
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 -
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,
Therefore, co-ordinates of the centroid (
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
Therefore coordinates of centroid are
∴ 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
Also, in Δ ABD
Therefore, Δ ABD becomes an equilateral triangle
So, AD = 2 units
Now, applying distance formula between
Squaring both sides
Again applying distance formula between
Squaring both sides
Therefore, coordinates of )
Note- ) is neglected because in the problem
Now, the centroid of
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
(x2,y2) = (-2,0)
(x3,y3) = (p,q)
We know that, Centroid of a triangle is given by
∴ 5 ∴
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.