Centroid of a Triangle

The Centroid of a triangle is the point of intersection of the medians of a triangle.

To find the centroid of a triangle

Let A (x1, y1), B (x2, y2) and C (x3, y3) are  the three vertices of the ∆ABC .

Let D be the midpoint of side BC.

Since, the coordinates of B (x2, y2) and C (x3, y3), the coordinate of the point D are (x2+x32, y2+y32).

Let G(x, y) be the centroid of the triangle ABC.

Then, from the geometry, G is on the median AD and it divides AD in the ratio 2 : 1, that is AG : GD = 2 : 1.

Therefore, x = {2(x2+x3)2+1x12+1} = x1+x2+x33

y = {2(y2+y3)2+1y12+1} = y1+y2+y33

Therefore, the coordinate of the G are (x1+x2+x33, y1+y2+y33)

Hence, the centroid of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) has the coordinates (x1+x2+x33, y1+y2+y33).


Note: The centroid of a triangle divides each median in the ratio 2 : 1 (vertex to base).


Solved examples to find the centroid of a triangle:

1. Find the co-ordinates of the point of intersection of the medians of trangle ABC; given A = (-2, 3), B = (6, 7) and C = (4, 1).

Solution:

Here, (x1  = -2, y1 = 3), (x2  = 6, y2 = 7) and  (x3  = 4, y3 = 1),

Let G (x, y) be the centroid of the triangle ABC. Then,

x = x1+x2+x33 = (2)+6+43 = 83

y = y1+y2+y33 = 3+7+13 = 113

Therefore, the coordinates of the centroid G of the triangle ABC are (83, 113)

Thus, the coordinates of the point of intersection of the medians of triangle are (83, 113).


2. The three vertices of the triangle ABC are (1, -4), (-2, 2) and (4, 5) respectively. Find the centroid and the length of the median through the vertex A.

Solution:

 Here, (x1  = 1, y1 = -4), (x2  = -2, y2 = 2) and  (x3  = 4, y3 = 5),

Let G (x, y) be the centroid of the triangle ABC. Then,

x = x1+x2+x33 = 1+(2)+43 = 33 = 1

y = y1+y2+y33 = (4)+2+53 = 33 = 1

Therefore, the coordinates of the centroid G of the triangle ABC are (1, 1).

D is the middle point of the side BC of the triangle ABC.

Therefore, the coordinates of D are ((2)+42, 2+52) = (1, 72)

Therefore, the length of the median AD = (11)2+(472)2 = 152 units.


3. Two vertices of a triangle are (1, 4) and (3, 1). If the centroid of the triangle is the origin, find the third vertex.

Solution:

Let the coordinates of the third vertex are (h, k).

Therefore, the coordinates of the centroid of the triangle (1+3+h3, 4+1+k3)

According to the problem we know that the centroid of the given triangle is (0, 0)

Therefore,

1+3+h3 = 0 and 4+1+k3 = 0

⟹ h = -4 and k = -5

Therefore, the third vertex of the given triangle are (-4, -5).

 Distance and Section Formulae





10th Grade Math

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