Centroid

Definition

The centroid is the centre point of the object. In mathematical terms, the point at which the three medians of the triangle intersect is known as the centroid of a triangle. Let us recall what we mean by a median of a triangle? The median is a line that joins the midpoint of a side and the opposite vertex of the triangle. It is important to understand here that the centroid of the triangle separates the median in the ratio of 2: 1.

Properties of a centroid

A centroid has the following properties –

A centroid is the centre of an object, such as a triangle or a square.
The centroid of a triangle is the point of intersection of all the three medians of a triangle.
A centroid always lies inside the object.
The centroid of a triangle divides the medians in the ratio 2 : 1

Formula for Centroid of a Triangle

Let us now understand the formula of the centroid of a triangle. For this purpose, let there be a triangle ABC, with the vertices a ( x ₁ , y ₁ ) , B ( x ₂, y ₂ ) and C ( x ₃, y ₃ ). Then for calculating the centroid of this triangle, we will take the average of the X and Y coordinates of all the three vertices. Hence, the centroid of the triangle ABC will be –

The centroid of the triangle = ( $\frac{x_1+ x_2+ x_3}{3}$ , $\frac{y_1+ y_2+ y_3}{3}$ )

Hence, we can say that for a triangle ABC, with the vertices a ( x ₁ , y ₁ ) , B ( x ₂, y ₂ ) and C ( x ₃, y ₃ ), the centroid will be given by ( $\frac{x_1+ x_2+ x_3}{3}$ , $\frac{y_1+ y_2+ y_3}{3}$ ) .

Let us understand the centroid through an example.

Example

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

Solution

We have been given a triangle whose vertices are A( 2, 6 ), B( 4, 9 ), and C( 6,15 ). We need to find the centroid of this triangle.

Now, we know that for a triangle ABC, with the vertices a ( x ₁ , y ₁ ) , B ( x ₂, y ₂ ) and C ( x ₃, y ₃ ), the centroid will be given by ( $\frac{x_1+ x_2+ x_3}{3}$ , $\frac{y_1+ y_2+ y_3}{3}$ )

Therefore, the centroid of the triangle having vertices A( 2, 6 ), B( 4, 9 ), and C( 6,15 ) will be

Centroid of the triangle = ( $\frac{( 2+4+6 )}{3}$ , $\frac{( 6+9+15 )}{3}$ ) = ( $\frac{12}{3}$ , $\frac{30}{3}$ ) = ( 4 , 10 )

Hence, the centroid of the triangle having vertices A( 2, 6 ), B( 4, 9 ), and C( 6,15 ) will be ( 4 , 10 )

Centroid Theorem

Let us now understand the centroid theorem. The centroid theorem states that the centroid of the triangle is at 2 / 3 of the distance from the vertex to the mid-point of the sides. Let us understand it through an example. Suppose we have, a triangle PQR, having a centroid at the point V.

Then, if S, T and U are the midpoints of the sides of the triangle PQ, QR and PR, respectively, according to the centroid theorem, we will have,

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

Similar to the centroid, we have another term in geometry, which is known as orthocentre. Let us see what the differences between the two are.

Difference between Orthocentre and Centroid

We know that the orthocentre is the point of intersection of the altitudes of a triangle. Recall that an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base ( the side opposite the vertex ). The following are the differences between an orthocentre and a centroid

Centroid	Orthocentre
The point in which the three medians of the triangle intersect is known as the centroid of a triangle.	The orthocentre is the point of intersection of the altitudes of a triangle.
The centroid of a triangle always lies inside the triangle	The orthocentre of a triangle may lie inside or outside of a triangle
The centroid of a triangle divides the medians of the triangle in the ratio 2 : 1	The orthocentre does not divide the altitudes of a triangle in any particular ratio.

Difference between Centroid and Incentre

We know that the incentre is the point of intersection of the angle bisectors of a triangle. Recall that the angle bisector of a triangle bisects the opposite side in such a way that the ratio of the two line segments is proportional to the ratio of the other two sides. Thus the relative lengths of the opposite side (divided by the angle bisector) are equated to the lengths of the other two sides of the triangle. The angle bisector theorem is applicable to all types of triangles.

The following are the differences between an incentre and a centroid

Centroid	Incentre
The point in which the three medians of the triangle intersect is known as the centroid of a triangle.	The incentre is the point of intersection of the angle bisectors of a triangle.
The centroid of a triangle divides the medians of the triangle in the ratio 2 : 1	The incentre does not divide the angle bisectors of a triangle in any particular ratio.

Key Facts and Summary

The centroid is the centre point of the object.
In mathematical terms, the point in which the three medians of the triangle intersect is known as the centroid of a triangle.
A centroid always lies inside the object.
The centroid of a triangle divides the medians in the ratio 2 : 1
For a triangle ABC, with the vertices a ( x ₁ , y ₁ ) , B ( x ₂, y ₂ ) and C ( x ₃, y ₃ ), the centroid will be given by ( $\frac{x_1+ x_2+ x_3}{3}$ , $\frac{y_1+ y_2+ y_3}{3}$ ) .
The centroid theorem states that the centroid of the triangle is at 2 / 3 of the distance from the vertex to the mid-point of the sides.
The orthocentre is the point of intersection of the altitudes of a triangle.
An altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base ( the side opposite the vertex ).
The incentre is the point of intersection of the angle bisectors of a triangle.
The angle bisector of a triangle bisects the opposite side in such a way that the ratio of the two line segments is proportional to the ratio of the other two sides.

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