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2015 AMC 10B Problems/Problem 13: Difference between revisions

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==Solution==
==Solution==
We find the x-intercepts and the y-intercepts to find the intersections of the axes and the line. If <math>x=0</math>, then <math>y=12</math>. If <math>y</math> is <math>0</math>, then <math>x=5</math>. Our three vertices are <math>(0,0)</math>, <math>(5,0)</math>, and <math>(0,12)</math>. Two of our altitudes are <math>5</math> and <math>12</math>. Since the area of the triangle is <math>30</math>, our final altitude has to be <math>30</math> divided by the hypotenuse. By the Pythagorean Theorem, our hypotenuse is <math>13</math>, so the sum of our altitudes is <math>\boxed{\textbf{(E)} \dfrac{281}{13}}</math>.
We find the x-intercepts and the y-intercepts to find the intersections of the axes and the line. If <math>x=0</math>, then <math>y=12</math>. If <math>y</math> is <math>0</math>, then <math>x=5</math>. Our three vertices are <math>(0,0)</math>, <math>(5,0)</math>, and <math>(0,12)</math>. Two of our altitudes are <math>5</math> and <math>12</math>. Since the area of the triangle is <math>30</math>, letting <math>h</math> be our hypotenuse, our final altitude has to be <math>30(2)/h</math>. By the Pythagorean Theorem, <math>h=13</math>, so the sum of our altitudes is <math>\boxed{\textbf{(E)} \dfrac{281}{13}}</math>.


==See Also==
==See Also==
{{AMC10 box|year=2015|ab=B|num-b=12|num-a=14}}
{{AMC10 box|year=2015|ab=B|num-b=12|num-a=14}}
{{MAA Notice}}
{{MAA Notice}}

Revision as of 13:20, 5 March 2015

Problem

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?

$\textbf{(A) } 20 \qquad\textbf{(B) } \dfrac{360}{17} \qquad\textbf{(C) } \dfrac{107}{5} \qquad\textbf{(D) } \dfrac{43}{2} \qquad\textbf{(E) } \dfrac{281}{13}$

Solution

We find the x-intercepts and the y-intercepts to find the intersections of the axes and the line. If $x=0$, then $y=12$. If $y$ is $0$, then $x=5$. Our three vertices are $(0,0)$, $(5,0)$, and $(0,12)$. Two of our altitudes are $5$ and $12$. Since the area of the triangle is $30$, letting $h$ be our hypotenuse, our final altitude has to be $30(2)/h$. By the Pythagorean Theorem, $h=13$, so the sum of our altitudes is $\boxed{\textbf{(E)} \dfrac{281}{13}}$.

See Also

2015 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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