How to find the least path consisting of the segments AP, PQ and QB Let $A = (0, 1)$ and $B = (2, 0)$ in the plane. 
Let $O$ be the origin and $C = (2, 1)$ .
Consider $P$ moves on the segment $OB$ and
$Q$ move on the segment $AC$.
Find the coordinates of $P$ and $Q$ for which the length of the path consisting of the segments $AP, PQ$ and Q$B$ is least.

Hint: Let $A'$ be the point one unit above $A$.
Let $B'$ be the point one unit below $B$.
Join $A'$ and $B'$ by a straight line. Show that gives the length of the minimal path.  
