Here you will learn formula for centroid of triangle and how to find centroid in a triangle with example.
Let’s begin –
Centroid in a Triangle
If A(\(x_1,y_1\)), B(\(x_2,y_2\)) and C(\(x_3,y_3\)) are vertices of any triangle ABC, then
The centroid is the point of the intersection of the medians(line joining the mid-point of sides and opposite vertices).
Centroid divides each median in the ratio of 2 : 1.
Formula for Centroid
The formula for centroid of a triangle is
(\(x_1+x_2+x_3\over 3\),\(y_1+y_2+y_3\over 3\))
where \(x_1\), \(x_2\), and \(x_3\) are x-coordinates of the vertices of the triangle; and \(y_1\), \(y_2\), and \(y_3\) are y-coordinates of the vertices of the triangle.
Example 1 : Find the centroid of a triangle whose vertices are (6,4), (3,1) and (1,2).
Solution : Given coordinates of triangle
\(x_1, y_1\) = (5,4)
\(x_2, y_2\) = (3,3)
\(x_3, y_3\) = (1,2)
centroid = (\(x_1+x_2+x_3\over 3\),\(y_1+y_2+y_3\over 3\))
= (\(5+3+1\over 3\), \(4+3+2\over 3\))
= (3,3)
Hence centroid is (3,3)
Example 2 : Find the centroid of a triangle whose vertices are (0,1), (2,0) and (-3,0).
Solution : Given coordinates of triangle
\(x_1, y_1\) = (0,1)
\(x_2, y_2\) = (2,0)
\(x_3, y_3\) = (-3,0)
centroid = (\(x_1+x_2+x_3\over 3\),\(y_1+y_2+y_3\over 3\))
= (\(0+2-3\over 3\), \(1+0+0\over 3\))
= (\(-1\over 3\), \(1\over 3\))
Hence centroid is (\(-1\over 3\), \(1\over 3\))
Condition for collinearity of three given points
Three given points A(\(x_1,y_1\)), B(\(x_2,y_2\)) and C(\(x_3,y_3\)) are collinear if any one of the following conditions are satisfied
(i) Area of triangle ABC is zero.
(ii) Slope of AB = Slope of BC = Slope of AC. i.e. \(y_2-y_1\over {x_2-x_1}\) = \(y_3-y_2\over {x_3-x_2}\) = \(y_3-y_1\over {x_3-x_1}\)
(iii) find the equation of line passing through 2 given points, if the third point satisfies the given equation of the line, then three points are collinear.
Hope you learnt formula of centroid. To learn more practice more questions to get ahead in competition. Good Luck!
