Derive an expression for the coordinates of a point that divides the line joining the points A(x1,y1,z1) and B(x2,y2,z2) internally in the ratio m:n .Hence find the coordinates of the midpoint of AB where A=(1,2,3) and B=(5,6,7).
Proof:From A,B, and R , draw perpendicular AL,BM and RN on the xy-plane.
Also draw AS perpendicular BM ,meeting BM and RN at S and T respectively.
From similar triangles ART and ABS ,we have
RT/BS = AR/AB
So RN-TN/BM - SM = m/m+n
And RN-AL/BM-AL = m/m+n
So z-z1/ z2 - z1 = m/m+n
So z = mz2 + nz1 /(m+n)
Similarly x = (mx2 +nx1)/ (m+n) and y = (my2 +ny1)/(m+n)
As for the mid-point m = n= 1
So x = 1+5 /2 = 3 , y = 2+6/2 =4 , z = 3+7/2 = 5
So the coordinate of mid-point are (3,4,5)