2015 AMC 12B Problem 11

Problem 11 of 25IntermediateGeometry

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?

Show solution

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.