2013 Mathcounts State Prep : Inscribed Circle Radius and Circumscribed Circle Radius of a right triangle

Question: $\mathrm{\Delta }$ ABC is a right triangle and a, b, c are three sides, c being the hypotenuse.
What is a. the radius of the inscribed circle and
b. the radius of the circumscribed circle? 

Solution a :
Area of the right $\mathrm{\Delta }$ABC =  ${\displaystyle \frac{ab}{2}}$ = ${\displaystyle \frac{(a+b+c)\times r}{2}}$
r =${\displaystyle \frac{ab}{a+b+c}}$

Solution b: 
In any right triangle, the circumscribed diameter is the same as the hypotenuse, so the circumscribed radius is${\displaystyle \frac{1}{2}}$ of the hypotenuse, in this case ${\displaystyle \frac{1}{2}}$ of c or ${\displaystyle \frac{1}{2}}$ of $\overline{AC}$



Some other observations: 
A. If you only know what the three vertices of the right triangle are on a Cartesian plane, you can use distance formula to get each side length and from there find the radius.

B.In right $\mathrm{\Delta }$ABC , $\overline{AC}$ is the hypotenuse.
If you connect B to the median of $\overline{AC}$, then $\overline{BD}$ = $\overline{AD}$ = $\overline{CD}$ = radius of the circumscribed circle