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