Question

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?

$\dfrac{3\sqrt{5}}{2}$

$\dfrac{7}{2}$

$\sqrt{15}$

$\dfrac{\sqrt{65}}{2}$

$\dfrac{9}{2}$

Answer: D

Concepts:incircle, incenter, and inradius circumcircle, circumcenter, and circumradius right triangle coordinate geometry

Difficulty rating: 1770

Solution:

Since $5^2 + 12^2 = 13^2,$ the triangle is right. Place it at $(0, 0),$ $(5, 0),$ $(0, 12).$ The circumcenter is the midpoint of the hypotenuse, $\left(\tfrac52, 6\right).$

The inradius satisfies $(12 - r) + (5 - r) = 13,$ so $r = 2$ and the incenter is $(2, 2).$

The distance is $\begin{gathered} \sqrt{\left(\tfrac52 - 2\right)^2 + (6 - 2)^2} \\ = \sqrt{\tfrac14 + 16} \\ = \dfrac{\sqrt{65}}{2}. \end{gathered}$

Thus, the correct answer is D.

2004 AMC 10B Problem 22

Answer: D

Difficulty rating: 1770

Solution:

Problem 22 in Other Years