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?

$20$

$\dfrac{360}{17}$

$\dfrac{107}{5}$

$\dfrac{43}{2}$

$\dfrac{281}{13}$

Answer: E

Concepts:coordinate geometry right triangle altitude

Difficulty rating: 1510

Solution:

The line meets the axes at $(5, 0)$ and $(0, 12),$ so the triangle is right with legs $5$ and $12$ and hypotenuse $13.$ Its area is $30.$

Two altitudes are the legs $5$ and $12;$ the altitude to the hypotenuse is $\dfrac{2 \cdot 30}{13} = \dfrac{60}{13}.$ The sum is $17 + \dfrac{60}{13} = \dfrac{281}{13}.$

Thus, the correct answer is E.

2015 AMC 12B Problem 11

Answer: E

Difficulty rating: 1510

Solution:

Problem 11 in Other Years