SOLUTION: find the radius of the circle inscribed in a triangle with lengths of 12, 12 and 8

Algebra ->  Circles -> SOLUTION: find the radius of the circle inscribed in a triangle with lengths of 12, 12 and 8      Log On


   

Question 596742: find the radius of the circle inscribed in a triangle with lengths of 12, 12 and 8
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The center, O, of the inscribed circle is at the intersection of the angle bisectors AD and CO, which is at the same distance from the sides of the triangle.
DO = EO is the radius of the circle.
We can use the given measures of the triangle sides and a little trigonometry to find radius DO
AB=AC=12 and AD=DC=AC%2F2=8%2F2=4
cos%28BCA%29=4%2F12=1%2F3
sin%28BCA%29=sqrt%281-%281%2F3%29%5E2%29=sqrt%281-1%2F9%29=sqrt%288%2F9%29=2sqrt%282%29%2F3
Angles DCO and ECO are congruent and their measure is half the measure of angle BCA.
Using the trigonometric identity for half angles
tan%28theta%2F2%29=sin%28theta%29%2F%281%2Bcos%28theta%29%29
we can calculate tan%28DCO%29=DO%2FDC=DO%2F4 and from there we can find radius DO.

DO%2F4=sqrt%282%29%2F2 --> highlight%28DO=2sqrt%282%29%29