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

Question: $\Delta$ 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 $\Delta$ABC =  $\dfrac {ab} {2}$ = $\dfrac {\left( a+b+c\right) \times r} {2}$
r =$\dfrac {ab} {a+b+c}$

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