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Question

The points A (x1,y1),B(x2,y2) and C(x3,y3) are the vertices of Δ ABC.
The median AD meets BC at D.
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.

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

The correct option is C x1+x2+x33,y1+y2+y33
The points A(x1,y1),B(x2,y2) and C(x3y3) are the vertices of ΔABC.
the median BE from B will meet AC at E which is the mid point of AC.
So, by section formula for mid point, the co-ordinates of E is E(x1+x32,y1+y32)=(x4,y4) .
the co-ordinates of Q(x,y) are by the section formula,
x=mx1+nx4m+n=x1+x32×2+x2×12+1=x1+x2+x33 and y=my1+ny4m+n=y1+y32×2+y2×12+1=y1+y2+y33
Q(x,y)=(x1+x2+x33,y1+y2+y33).
Proceeding in the same way the co-ordinates of R(p,q), which divides the median CF in the ratio 2:1, are
p=x1+x22×2+x3×12+1=x1+x2+x33 and
q=y1+y22×2+y3×12+1=y1+y2+y33
R(p,q)=(x1+x2+x33,y1+y2+y33)
So, Q and R are same points.
Q(x,y)=R(p,q)=(x1+x2+x33,y1+y2+y33)

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