2015 AMC 10B Problem 13

The triangle is a right triangle with legs of $12$ and $5.$ This makes the hypotenuse $13.$ Two of the altitudes are then $12$ and $5.$ Also, for any side, $A = \frac{bh}2$ where $b$ is the base and $h$ is the altitude. The area is $\frac {12\cdot 5}2 = 30,$ so the other altitude $h$ can be found with $30 = \frac{13h}2.$ Thus, this altitude is $\frac{60}{13}.$ Therefore, the sum is $$12+5+\dfrac{60}{13} = \dfrac{281}{13} .$$ Thus, the correct answer is E .