[Maths Class Notes] on Centroid Pdf for Exam

When we draw from the vertices of a triangle to the centre of the opposite sides from the centroid of the right-angle triangle, the opposite point of the three medians is called the centroid of right-angle triangles.

Relationships Between Orthocentre, Centroid and Incentre

• An orthocentre is a place where three triangular altitudes meet. The part of the line formed from one vertex to the other, perpendicular to the other, is known as the altitude of the triangle.

• The centroid as it is known is the intersection point of three medians. The median is where each straight line connects the middle part of the side with the opposite vertex.

• The incentre is the connecting point of the three perpendicular bisectors. The angle-bisector triangle lines are drawn directly from the centre of the triangle.

• The Centroid triangle divides the line connecting the incentre and the orthocentre by a ratio of 1: 2.

• Think of H, O and G as the centre of the orthocentre, incentre and centroid of any triangle. Here, G divides part of the OH line starting at O ​​with a ratio of 1: 2, inside, that is, OG / GH = 1: 2

Concluding, Orthocentre, incentre, and centroid are always binding in a straight line, known as the Euler line. The centroid is always centred between an orthocentre and a triangle incentre. In the equal triangle, the orthocentre, the incentre, and the centroid, all lie in the same position, within the triangle. In the obtuse triangle, orthocentre, incentre, both lie outside the triangle and the centroid is inside the triangle.

The centroid is the centre of the object. Each shape such as triangle, circle, trapezium, square, semi-circle, etc. has a centroid of it. It helps to establish a centre of gravity. In geometry, the centroid of the triangle is much more focused. The triangle has three mediums and the centroid is the place where all medians meet. The centroid triangle divides the median in a 2: 1 ratio where the distance between vertex and centroid is twice the distance between centroid and midpoint.

Explanation of the Centroid Theorem

The centroid theorem states that in a triangle, the centroid is at 2/3 of the distance from the vertex to the midpoint of the sides.

()

Let us understand the centroid theorem with an example by considering a triangle ABC with centroid M. D, E, and F are the midpoints of the sides BC, AC, and AB, respectively. By applying the centroid theorem, we get – AM = 2/3AD, BM = 2/3BE, and CM = 2/3CF.

Centroid Properties and Formula

Following are the properties of the centroid:

• It is defined as the centre of the object.

• The centroid should always lie inside the object.

• It is also the centre of gravity.

• The centroid is the point of concurrency of all the medians. 

Now, let us learn the centroid formula by considering a triangle. Suppose that the three vertices of the triangle are given by the coordinates, A(x1, y1), B(x2, y2), and C(x3, y3), as shown in the figure below. Then, we can calculate the centroid of the triangle by taking the average of the x coordinates and the y coordinates of all the three vertices. So, the centroid formula can be mathematically expressed as G(x, y) = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3).

()

Solved Examples on Centroid of the Triangle

Question 1:

The vertices of a triangle are A(4, 9), B(6, 15), and C(2, 6). Find its centroid.

( )

Answer :

The coordinates of the three vertices of the triangle, ABC are as follows:

A(x1, y1) = A(4, 9)

B(x2, y2) = B(6, 15)

C(x3, y3) = C(2, 6)

Now, we have to find the centroid of the triangle ABC. We know that the formula for finding the centroid of the triangle is given by – ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3).

So, let us substitute the corresponding values in this formula and get the resultant centroid. 

The centroid of the triangle ABC = ((4 + 6 + 2)/3, (9 + 15 + 6)/3) = (12/3, 30/3) = (4, 10)

Hence, the centroid of the triangle ABC with vertices A(4, 9), B(6, 15), and C(2, 6) is (4, 10).

Question 2: 

In the figure given below, C is the centroid of the triangle RST. If RE = 21, find RC. 

 

()

Answer :

We can find the solution to this question in two ways. 

Method 1

We know that the centroid of a triangle is at 2/3 of the distance from the vertex to the midpoint of the sides.

It implies that RC = 2/3RE

So, RC = 2/3 (21)

RC = 2 * 7 = 14

Method 2

We know that the centroid of the triangle divides all its medians in the ratio 2:1.

So, RC = 2x and CE = x

RC + CE = RE 

2x + x = 21

3x = 21

x = 21/3

x = 7

RC = 2x = 2*7 = 14

Hence, the value of RC is 14. 

Question 3:

The vertices of a triangle PQR are given as P(2, 1), Q(a, 2), and R(-2, b), and its centroid is (1, 7/3). Find the value of a and b. 

Answer :

The coordinates of the three vertices of the triangle, PQR are as follows:

P(x1, y1) = P(2, 1)

Q(x2, y2) = Q(a, 2)

R(x3, y3) = R(-2, b)

The centroid, let us say, O = (1, 7/3)

For finding the value of a and b, we will use the centroid formula of the triangle, which is given by – ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3).

Now, let us substitute the given values and find the value of a and b.

(1, 7/3) = ((2 + a + (-2)/3), (1 + 2 + b)/3) 

(1, 7/3) = ((a/3), (3 + b)/3)

a/3 = 1 

a = 3

and (3 + b)/3) = 7/3

3 + b = 7

b = 4

Hence, the value of a is 3 and b is 4.

< span>Centroid Formula for Different Shapes

()