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Question

The points A (x1,y1),B(x2,y2) and C(x3y3) are the vertices of Δ ABC.
The median AD meets BC at D.
What are the coordinates of the centroid of the triangle ABC?

A
2x1+2x2+x33,2y1+2y2+y33
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B
x1+x2+x33,y1+y2+y33
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C
x1+x2+x33,y1+y2+y33
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D
x1+x2+2x33,y1+y2+2y33
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Solution

The correct option is B x1+x2+x33,y1+y2+y33
The points A(x1,y1),B(x2,y2) and C(x3y3) are the vertices of ΔABC.
Let the median AD meet BC at D.
To find: Co-ordinates of its centroid G(X,Y) .

G(X,Y) of ΔABC will divide its median AD in the ratio m:n=2:1
i.e AG:GD=2:1 when D(x4,y4) is the mid-point of BC.
So, the co-ordinates of D(x4,y4), by the section formula for mid point, are
D(x4,y4)=(x3+x22,y3+y22).
the co-ordinates of G(X,Y) by the section formula is,

X=nx1+nx4m+n=x1+x32×2+x2×12+1=x1+x2+x33 and

Y=ny1+ny4m+n=y1+y32×2+y2×12+1=y1+y2+y33

G(X,Y)=(x1+x2+x33,y1+y2+y33)

The co-ordinates of the centroid G(X,Y)=(x1+x2+x33,y1+y2+y33)

140862_81269_ans.png

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