Question 118439
Plot the points P and Q on a graph.  Then construct the vertical line x = 1.  R can then be any point on that vertical line.


If R is any point that is NOT on line PQ, then PQR forms a triangle and PR + RQ  > PQ.  But, if R is colinear with P and Q, then PR + RQ = PQ.  Lastly, there is no possible value for m that would create a situation where PR + RQ < PQ.  Therefore, the minimum value for m is the point where R is colinear with P and Q.


Now all you have to do is find the equation for the line containing P and Q and evaluate it at x = 1 to find your value for m.  I assume that you know the two-point form of a straight line, and leave the rest to you.  If you are still having trouble, write back and we'll work it out.


{{{drawing(600,600,-3,5,-3,3,
grid(0.2),
green(locate(-.9,-2.1,P(-1,-2))),
red(circle(-1,-2,.05)),
green(locate(4.1,1.9,Q(4,2))),
red(circle(4,2,.05)),
line(-5,-(26/5),5,(14/5)),
blue(line(1,-5,1,5)),
blue(circle(1,1.5,.05)),
blue(locate(1.1,1.4,R(1,m))),
green(line(1,1.5,4,2)),
green(line(1,1.5,-1,-2))
)}}}


Hope that helps,
John