Let A(4,2), B(6,5) and C(1,4) be the vertices of `Delta ABC `
(1) The median from A meets BC at D. Find the coordinates of the points D.
(2) Find the coordinates of the points P on AD such that AP : PD = 2 : 1
(3) Find the coordinates of points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RE = 2 : 1
(4) What do you observe ?
[Note : The point which is common to all the three medians is called the centroid and this point divides each median in the ratio 2 : 1 ]
(5) If `A(x_(1), y_(1)), B(x_(2), y_(2)) and C(x_(3),y_(3))` are the vertices of `Delta ABC` find the coordinates of the centroid of the triangle

Answer

(1) `((7)/(2),(9)/(2))`
(2) `((11)/(3),(11)/(3))`
(3) `(5,(7)/(2))`
(4) P = Q = R
Centroid of a triangle divides each of its medians in the ratio 2 : 1 from the vertex side
(5) `((x_(1) + x_(2) + x_(3))/(3) , (y_(1) + y_(2) + y_(3))/(3))`