CENTROID OF THE TRIANGLE

The 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

Examples of Finding Centroid of the Triangle Formed by the points

Example 1 :

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

Solution :

Centroid of a triangle = (x+ x+ x3)/3, (y+ y+ y3)/3

Here we have x1 = 8, x2 = 1 and x3 = 3

y1 = 4, y2 = 3 and y3 = -1

=  (8 + 2 + 3)/3, (4 + 3 -1)/3

=  (12/3), (6/3)

=  (4, 2)

Therefore the centroid is (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(a, b) be the required vertex. So the vertices of the triangle are (2, -1) (7, 8) and (a, b) and the centroid is (4, 3)

Here we have x1  =  2, x2  =  7 and x3  =  a

y1  =  -1, y2  =  7 and y3  =  b

By applying the formula

Centroid of a triangle = (x+ x+ x3)/3, (y+ y+ y3)/3

 (2 + 7 + a)/3  =  49 + a  =  12a  =  12 - 9a  =  3 (-1 + 8 + b)/3  =  37 + b  =  9b  =  9 -7b  =  2

Hence the required vertex is (3, 2).

After having gone through the stuff given above, we hope that the students would have understood how to find the coordinates of the centroid of a triangle with vertices.

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