Coordinate System and Coordinates
10.0 Centroid of a triangle
10.0 Centroid of a triangle
The point of intersection of the medians of a triangle is called the centroid of a triangle and it divides the median internally in the ratio $2:1$.
If $A({x_1},{y_1})$, $B({x_2},{y_2})$ and $C({x_3},{y_3})$ are the vertices of a $\Delta ABC$, whose sides $BC$, $CA$ and $AB$ are of lengths $a$, $b$ and $c$ respectively, then co-ordinates of centroid of $\Delta ABC$ is $$G \equiv (\frac{{{x_1} + {x_2} + {x_3}}}{3},\frac{{{y_1} + {y_2} + {y_3}}}{3})$$
Note:
- If $({a_1},{b_1})$, $({a_2},{b_2})$, $({a_3},{b_3})$ are the mid-points of the sides of a triangle, then its centroid is given by $$(\frac{{{a_1} + {a_2} + {a_3}}}{3},\frac{{{b_1} + {b_2} + {b_3}}}{3})$$
- From figure, Area of $\Delta AGB$ = Area of $\Delta AGC$ = Area of $\Delta BGC$ = $\frac{1}{3}$ Area of $\Delta ABC$
