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Question

The points A x1,y1 , Bx2,y2 , Cx3,y3 are the vertices of 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 points of coordinates Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1.
(iv) What are the coordinates of the centropid of the triangle ABC ?

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Solution

(i) Median AD of the triangle will divide the side BC in two equal parts.



Therefore, D is the midpoint of side BC.
Coordinates of D are
x2+x32,y2+y32

(ii)
THe point P divided the side AD in the ratio 2: 1.
Coordinates of P are
2×x2+x32+1×x12+1, 2×y2+y32+1×y12+1=x1+x2+x33,y1+y2+y33


(iii)
Median BE of the triangle will divide the side AC in two equal parts.
Therefore, E is the midpoint of side AC.
Coordinates of E are
x1+x32,y1+y32
The point Q divided the side BE in the ratio 2: 1.
Coordinates of Q are
2×x1+x32+1×x22+1, 2×y1+y32+1×y22+1=x1+x2+x33,y1+y2+y33

Similarly, Coordinates of Q are R are x1+x2+x33,y1+y2+y33.

(iv)
The points P, Q and R coincides and is the centroid of the triangle ABC.
So, coordinates of the centroid is x1+x2+x33,y1+y2+y33.

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