  Question

# Question 3 (iii) The points A(x1,y1),B(x2,y2) and C(x3,y3) are the vertices of Δ ABC. 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.

Solution

## Let the coordinates of the points Q be (p,q) and R be (r,s) ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣Since, BE is the median of side CA, so BE divides AC into two equal parts∴mid−point of AC=coordinate of E⇒ E=(x1+x32,y1+y32)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣Since, CF is the median of side AB, so CF divides AB into two equal parts∴mid−point of AB=coordinate of F⇒ F=(x1+x22,y1+y22)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ Given: The point Q(p,q), divide the line joining B(x2,y2) and E(x1+x32,y1+y32) (as E is the mid point of AC) in the ratio 2:1. and  The point R(r,s), divide the line joining C(x3,y3) and F(x1+x22,y1+y22) (as F is the mid point of AB) in the ratio 2:1. The coordinates of Q Q≡⎛⎜⎝2.(x1+x32+1.x2)2+1,2.(y1+y32+1.y2)2+1⎞⎟⎠=Q(x1+x2+x33,y1+y2+y33) The coordinate of R R≡⎛⎜⎝2.(x1+x22+1.x3)2+1,2.(y1+y22+1.y3)2+1⎞⎟⎠=R(x1+x2+x33,y1+y2+y33) So, the required coordinate of points R and Q=(x1+x2+x33,y1+y2+y33) Hence the points Q and R are the same point as they have the same co-ordinates. Here it is the "centroid" as it is the point of intersection of medians  Suggest corrections   