Centroid of a Triangle


The simple definition for centroid of a triangle is, the point of concurrency of the medians.

In the above triangle , AD, BE and CF are called medians. All the three medians AD, BE and CF are intersecting at G. So  G is called centroid of the triangle

If the coordinates of A,B and C are (x1y1), (x2,y2), and (x3,y3) , then the formula to determine the centroid of the triangle given by

Example:1

Find the centroid of the triangle whose vertices are the points (8,4), (1,3) and (3,-1)

Solution:


The centroid of the triangle can be found by using the above formula.

Here, we have

(x1,y1) = (8,4)

(x2,y2) = (1,3)

(x3,y3) = (3,-1)

Plug the above values in to the formula


Centroid of the triangle = [(8+1+3)/3 , (4+3-1)/3]

= [12/3 , 6/3)

= (4 , 2)



Example:2

If a triangle has its centroid at (4,3) and two of its vertices are (2,-1) and (7,8). Find the third vertex.


Solution:


Let the third vertex be (a,b)

Here, we have

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

(x2,y2) = (7,8)

(x3,y3) = (a,b)

Plug the above values in to the formula

Centroid of the triangle = (4,3)

[(2+7+a)/3 , (-1+8+b)/3] = (4 ,3)

[(9+a)/3 , (7+b)/3] = (4 , 3)

Equating the coordinates of x and y , we are getting

(9+a)/3 = 4 , (7+b)/3 = 3

9+a = 4(3) , 7+b = 3(3)

9+a = 12 , 7+b = 9

a = 12- 9 , b = 9-7

a = 3 , b= 2

Hence the third vertex (a , b) = (3 , 2)


This is how we are using the above formula to find centroid of a triangle and If centroid of the triangle and two of the vertices are given , we can find the third vertex.




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