What is the centroid of the triangle with vertices (6, 4), ( 3,1) and (0, 4)?  

The centroid of a triangle is the point where the medians of the triangle meet. So we can find the centroid by determining the point of contact of two medians.

In the given triangle, the midpoint of (6, 4) and ( 3,1) is [( 6 + 3)/2 , (4 + 1)/2]. The equation of the line joining (0, 4) and ( 9/2 , 5/2) is

y – 4 = [( 4 – 5/2)/(0-9/2)]*(x – 0)

=> y – 4 = [(3/2) / (-9/2)]( x – 0)

=> -3 ( y – 4) = x

=> x + 3y – 12 = 0

The midpoint of ( 3,1) and (0, 4) is ( 3/2 , 5/2)

The line joining (6 , 4) and ( 3/2 , 5/2) is

y – 4 = [( 4 – 5/2)/(6 – 3/2)]( x – 6)

=> y – 4 = [( 3/2)/(9/2)]( x – 6)

=> 3(y – 4) = ( x – 6)

=> x - 3y + 6 = 0

We now need the point of intersection of x + 3y – 12 = 0 and x - 3y + 6 = 0.

Adding the two equations: 2x  - 6 = 0 or x = 3 and substituting x = 3 in x + 3y - 12 = 0 gives y = 3.

The centroid of the traingle is (3 , 3)