2009 AMC 12B Problem 13

Let $D$ be the foot of the altitude from $A.$ Then $BD = \sqrt{13^2 - 12^2} = 5$ and $DC = \sqrt{15^2 - 12^2} = 9.$ If $D$ lies between $B$ and $C,$ then $BC = 5 + 9 = 14$; if the triangle is obtuse, $BC = 9 - 5 = 4.$ The sum is $14 + 4 = 18.$ Thus, the correct answer is D.