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?

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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.

Tagged: coordinate geometry · right triangle · altitude