Thursday, February 21, 2013

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