Let A (4, 2), B(6, 5) and C (1, 4) be the vertices of Δ ABC
(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?
[Note: The point which is common to all the three medians is called the centroidand this point divides each median in the ratio 2: 1]
(v) If 
and 
are the vertices of Δ ABC, find the coordinates of the centroid of the triangle
(i) Median AD of the triangle will divide the side BC in two equal parts
Therefore, D is the mid-point of side BC
Coordinates of D = (
)
= (
)
(ii) Point P divides the side AD in a ratio 2:1
Coordinates of P = (
)
= (
(iii) Median BE of the triangle will divide the side AC in two equal parts.
Therefore, E is the mid-point of side AC
Coordinates of E = (
, 
)
= (
)
Point Q divides the side BE in a ratio 2:1
Coordinates of Q = (
)
= (
Median CF of the triangle will divide the side AB in two equal parts. Therefore, F is the mid-point of side AB
Coordinates of F = (
)
= (5, 
)
Point R divides the side CF in a ratio 2:1
Coordinates of R = (
)
= (
)
(iv) It can be observed that the coordinates of point P, Q, R are the same.Therefore, all these are representing the same point on the plane i.e., the centroid of the triangle
(v) Consider a triangle, ΔABC, having its vertices as A(x1, y1), B(x2, y2), and C(x3,y3)
Median AD of the triangle will divide the side BC in two equal parts. Therefore, D is the mid-point of side BC
Coordinates of D = (
, 
)
Let the centroid of this triangle be O.Point O divides the side AD in a ratio 2:1
Coordinates of O = (
)
= (
)