Question

The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?

$20$

$\dfrac{360}{17}$

$\dfrac{107}{5}$

$\dfrac{43}{2}$

$\dfrac{281}{13}$

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 .