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-z

_{1}/ z

_{2}- z

_{1}= m/m+n

So z = mz

_{2}+ nz

_{1}/(m+n)

Similarly x = (mx

_{2}+nx

_{1})/ (m+n) and y = (my

_{2}+ny

_{1})/(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)

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