¶ 2004 AMC 10B Problem 22
Problem:
A triangle with sides of , and has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The triangle is a right triangle that can be placed in a coordinate system with vertices at , and . The center of the circumscribed circle is the midpoint of the hypotenuse, which is .
To determine the radius of the inscribed circle notice that the hypotenuse of the triangle is
So the center of the inscribed circle is , and the distance between the two centers is
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions