## Introduction to Centroid Formula

The centroid is a very important topic of a triangle in geometry. The centroid is a geometrical center that is present in a two-dimensional plane surface. It is a point that is located at the center or the mean position of all the points present on the plane surface.

Based on sides and angles, a triangle is of six different types such as equilateral triangle, scalene triangle, isosceles triangle, acute-angled triangle, right-angled triangle, and obtuse-angled triangle. Let us find out the concept behind the centroid in these geometry shapes along with its concepts, theorem, properties, and centroid formula.

## What are the Centroid and the Centroid Theorem?

Simply we can say that the center point of a plane is known as centroid. The point where the three medians of the triangle intersect is known as the centroid of the triangle. The median is defined as the lines from each vertex of the triangle that joins the midpoint. The centroid of the triangle divides the median in a 2:1 ratio. The centroid of a triangle is calculated by taking the average of the X-coordinates and Y-coordinates of the triangle's vertices.

Similarly, the centroid theorem states that the centroid is at a two-thirds distance from the vertex and the sides’ midpoint in a triangle.

Let us consider a triangle ABC with the centroid D. Then the midpoints of the sides AB, BC, and AC are X, Y, and Z, respectively.

By applying the centroid theorem,

AD = 2/3 AX

BD = 2/3 AY

CD = 2/3 AZ

## Centroid Properties and Formula of Centroid

The properties of the centroid are:

It is called the center of the object.

It is also known as the center of gravity.

It lies inside the object.

The centroid is the concurrency point of all the medians.

Now let us learn the centroid of a triangle formula by considering a triangle.

Let's say there are 3 vertices in a triangle named by the coordinates A(X_{1}, Y_{1}), B(X_{2}, Y_{2}), and C(X_{3}, Y_{3}). The centroid of the triangle is considered the average of the three vertices’ X and Y coordinates.

So the formula of centroid of a triangle:

Z(X, Y) = {(X_{1} + X_{2} + X_{3})/3 , (Y_{1} + Y_{2} + Y_{3})/3}

Where Z is the centroid of the triangle.

**X _{1}, X_{2}, X_{3} -** X coordinates of vertices of triangle.

**Y _{1}, Y_{2}, Y_{3} -** Y coordinates of vertices of triangle.

### Conclusion

Here we learned about the centroid of the triangle which is an important portion in the geometry. The centroid of any closed geometrical shape always denotes the center of that shape. The centroid location is denoted by "D" with the X and Y coordinates. Centroid will always live inside the object and will be known as a center of gravity where all the points from the center will remain equal.

1. Derive the Formula of Centroid of a Triangle.

Ans: Let ABC be the vortex points of a triangle whose coordinates are A(X_{1}, Y_{1}), B(X_{2}, Y_{2}), and C(X_{3}, Y_{3}). The midpoints of the sides AB, BC, and AC are D, E, and F, respectively, and the centroid is denoted as K.

As D is the mid-point of the side AB, using the midpoint formula the coordinates of the

D = (X_{2} + X_{3})/2, (Y_{2} + Y_{3})/2

If the centroid K divides the median into a 2:1 ratio, then using the formula of centroid

X coordinates of K:

X = (X_{1} + X_{2} + X_{3})/3

Y = (Y_{1} + Y_{2} + Y_{3})/3

Show the coordinates of the formula of centroid of a triangle K is proved = K(X, Y) = {(X_{1} + X_{2} + X_{3})/3, (Y_{1} + Y_{2} + Y_{3})/3}

2. Using the Centroid of a Triangle Formula, Calculate the Centroid Whose Vertices are (8, 0), (2, 6), and (5, 4).

Ans: The given parameters are:

(X_{1}, Y_{1}) = (8, 0)

(X_{2}, Y_{2}) = (2, 6)

(X_{3}, Y_{3}) = (5, 4)

The formula of centroid of a triangle is given by:

K = {(X_{1} + X_{2} + X_{3})/3 , (Y_{1} + Y_{2} + Y_{3})/3}

= {(8 + 2 + 5)/3, (2 + 6 + 4)/3}

= {15/3, 12/3}

= 5, 4

3. If (2, 2) Coordinates are the Centroid of a Triangle Whose Vertices are (1, 4), (2, 0), (a, 2). Find the Value of p.

(X_{1}, Y_{1}) = (1, 4)

(X_{2}, Y_{2}) = (2, 0)

(X_{3}, Y_{3}) = (a, 2)

Centroid K = (2, 2)

Ans: Using the centroid formula

K(X, Y) = {(X_{1} + X_{2} + X_{3})/3 , (Y_{1} + Y_{2} + Y_{3})/3}

(2, 2) = {(1 + 2 + a)/3, (4 + 0 + 2)/3}

(2, 2) = (3a/3), (6/3)

Equating the X coordinates

3a/3 = 2

a = (2 × 3)/3

a = 3

Hence, the value of a = 3.