Let A(4,2), B(6,5) and C(1,4) be the vertices of ΔABC. (i) The median from A meets BC at D. Find the coordinates of the point D. (ii) Find the coordinates of the point P on AD such that AP:PD = 2:1 (iii) Find the coordinates of points Q and R on medians BE and CF respectively such that BQ:QE = 2:1 and CR:RF = 2:1. (iv) What do you observe?

i) Median is the line joining the midpoint of one side of a triangle to the opposite vertex. So, the coordinates of D would be (6+1/2, 5+4/2) = (7/2, 9/2)

ii) P divides AD in the ratio 2:1. A(x1,y1) = (4,2), D(x2,y2) = (7/2, 9/2) m:n = 2:1
Using section formula, we get the coordinates of P.
P(x,y) = (nx1 + mx2/m+n, ny1 + my2/m+n) = (14 + 27/2 / 2+1, 12 + 29/2 / 2+1) = (11/3, 11/3)

iii) Coordinates of E will be (5/2, 3) and the coordinates of F = (5, 7/2).
Coordinates of Q = (nx1 + mx2/m+n, ny1 + my2/m+n) = (16 + 25/2 / 2+1, 15 + 23/2 / 2+1) = (11/3, 11/3)
Coordinates of R = (nx1 + mx2/m+n, ny1 + my2/m+n) = (11 + 25/2 / 2+1, 14 + 27/2 / 2+1) = (11/3, 11/3)

iv) The coordinates of P, Q and R are the same which is (11/3, 11/3). This point is called the centroid, denoted by G.
v) Centroid of triangle ABC = (x1 + x2 + x3 / 3, y1 + y2 + y3 / 3)