A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?
+128475 Let A = (0,0)
Let B = (0,5)
Let C = (12,0)
AB = 5
AC = 12
BC = 13
We can calculate the incenter as
x = (0)(BC) + (0) (AC) + (12)(AB) 12 * 5
________________________ = ______ = 2
perimeter 30
y = (0)(BC) + (5)(AC) + (0)(AB) 5 * 12
_________________________ = __________ = 2
perimeter 30
So....the incenter is (2, 2)
The circumcenter occurs at the midpoint of the hypotenuse = [ (12 + 0) / 2 , (5 + 0) /2 ] =
( 6, 5/2) = (6, 2.5)
The distance between these points is
sqrt [ ( 6 - 2)^2 + ( 2.5 - 2)^2 ] = sqrt [ 16 + .25 ] = sqrt [16 + 1/4] = sqrt [65] / 2
