Maximize the inradius given the base and the area of the triangle BdMO 2013 Secondary:

A triangle has base of length 8 and area 12. What is the radius of the largest circle 
  that can be inscribed in this triangle?

Let $A,r,s$ denote the area,inradius and semiperimeter respectively.Then we have that
$A=rs$
$\implies 12=r\dfrac{8+b+c}{2}$  [$b,c$ are the lengths of two other sides]
The only thing that can be said about $b+c$ is that it must be greater than $8$.Also,to maximize $r$,we need to minimize $b+c$.Also,$b+c$ can be rewritten as $8+2y$ for some $y$.But I am unable to infer anything else from this information.A hint will be appreciated.

For b+c to be minimum b=c should hold. So b=c=5.(by simple Pythagoras). So r=1.34.
