Question

what are the coordinates of the centroid of each triangle given the three vertices of D(0,3),C(-1,0),B(4,0)?

Answer 1

The quickest, but little know method is to simply "average" the x's and the y's

x for centroid = (0-1+4)/3 = 1
y for centroid = (3+0+0)/3 = 1
the centroid is (1,1)

Slightly more complicated way:
find midpoint of DB = M(2 , 3/2)
join CM
Let the centroid be P(a,b)
we know that P must be such that CP : PM = 2:1
(a+1)/2 = (2-a)/1
a+1 = 4 - 2a
3a = 3
a = 1

(b-0)/2 = (3/2 - b)/1
b = 3 - 2b
3b=3
b = 1

centroid = P(a,b) = P(1,1)

most complicated way:

find the equation for CM
after finding the midpoint N of BC, find the equation for DN
intersect DN and CM
I will not do it, but am absolutely sure it will be (1,1)