Where is the centroid?

The solution to find the centroid of a triangle is the average of the coordinates of its vertices:

$\triangle ABC$ with vertices $A(x_1,y_1)$ , $B(x_2,y_2)$ , and $C(x_3, y_3)$ will have the centroid $\left(\dfrac{x_1 + x_2 + x_3}{3}, \dfrac{y_1 + y_2 + y_3}{3} \right)$ .

Using this generalization, we have $\left(\dfrac{2+8+9}{3}, \dfrac{4-2+9}{3}\right)=\left(\dfrac{19}{3},\dfrac{11}{3}\right) \implies a+b+c = 19+11+3=\boxed{33}$ .

For a triangle with co-ordinates (a,b) , (c,d) &
(e, f) , the co-ordinate of centroid g (x,y) is - g (x,y) = [(a+c+e)/3], [(b+d+f)/3]