Centroid

The centroid is indeed a crucial concept of a triangle (a polygon with three vertices, three edges, and three interior angles) in geometry. Based on the angles and sides, a triangle can be categorized into different types, such as equilateral triangle, isosceles triangle, scalene triangle, acute-angled triangle, obtuse-angled triangle, and right-angled triangle. Let us make ourselves familiar with the concept of the centroid for different geometric shapes in detail, along with the formula and properties.

Understanding the Term Centroid

The centre point of the object is what we refer to as the centroid. The point at which a triangle's three medians intersect is called the centroid of the triangle. We can also define the centroid as the point of intersection of the three medians. The median refers to the line joining the midpoint of a side to the opposite vertex of a triangle. The triangle’s centroid divides the median in the ratio 2:1. We can calculate the centroid by taking the average of the x-coordinates and the y-coordinates of the vertices of the triangle.

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.

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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).

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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.

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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. 

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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. 

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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.

Centroid Formula for Different Shapes

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