Centroid Right Angle Triangle
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.
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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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FAQs on Centroid
1. What do you mean by the centroid of a triangle?
We say that when three medians of a triangle are intersected then a centroid of a triangle is formed. Among the four concurrencies of a triangle centroid of a triangle is one such point. When a student joins the vertices of a triangle to the midpoint of the opposite sides of the triangle then that is termed as medians of a triangle. It is constructed when the other two are joined together. There are many figures attached in the book, the experts at Vedantu recommend you to analyze the diagrams well and understand what is the meaning.
2. What is the difference between the orthocentre and centroid of a triangle?
These are the three main points which the experts at Vedantu have distinguished. The following are the points of distinction:
1. The orthocentre consists of the point where all the altitudes intersect, whereas the centroid is the point where all medians intersect.
2. Orthocentre might exist outside the triangle whereas the centroid exists inside the triangle.
3. There is no perfect ratio in 2:1 which divides the altitudes in the orthocentre whereas the medians are divided into a 2:1 ratio in the centroid.
3. What is the difference between the incentre and centroid of a triangle?
There are differences between the incentre and centroid of a triangle based on which type the triangle is. The important differences according to the Vedantu experts are:
1. The intersection point of the angle bisectors is called the incentre whereas the intersection point of all medians is termed as a centroid.
2. Both the incentre and centroid of a triangle lie inside the triangle.
3. The angle bisectors are divided in the 2:1 whereas the centroid can be divided into a ratio of 2:1
4. Describe the easiest way of finding a centroid of a triangle?
The centroid of a triangle can be ascertained through three simple steps. The following are some basic steps that are to be followed to find the centroid of a triangle:
All the three coordinates of each vertex need to be identified and listed.
You need to add all the x-coordinates and divide the sum by 3 of all three vertices of a triangle.
You need to add all the y-coordinates and divide the sum by 3 of all three vertices of a triangle.
Hence, you have learned the easiest way of finding the centroid of a triangle.
5. Why choose Vedantu to download the resources of the centroid of a triangle?
Vedantu is the leading ed-tech company in India which has led to advancements in the field of study to some other level. The students have become familiar with the concepts and apprehensive questions curated by the Vedantu experts. The students can perfectly perceive the knowledge and concepts provided by the Vedantu experts. The Vedantu experts have always formulated questions keeping in mind the comprehensive power of the students. Everything is available free of cost on the Vedantu website.
