Difference between revisions of "2015 AMC 10B Problems/Problem 13"

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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>, and since it is a 5-12-13 right triangle, the hypotenuse is <math>13</math>. Since the area of the triangle is <math>30</math>, so our final altitude is <math>\frac{30(2)}{13}=\frac{60}{13}</math>. The sum of our altitudes is <math>\frac{60+156+65}{13}=\boxed{\textbf{(E)} \dfrac{281}{13}}</math>.
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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>, and since it is a 5-12-13 right triangle, the hypotenuse is <math>13</math>. Since the area of the triangle is <math>30</math>, so our final altitude is <math>\frac{30(2)}{13}=\frac{60}{13}</math>. The sum of our altitudes is <math>\frac{60+156+65}{13}=\boxed{\textbf{(E)} \dfrac{281}{13}}</math>. Note that there is no need to calculate the final answer after we know that the third altitude has length <math>\frac{60}{13}</math> since <math>E</math> is the only choice with a denominator of <math>13</math>.
  
 
==See Also==
 
==See Also==

Revision as of 18:01, 1 January 2018

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 $12$, and since it is a 5-12-13 right triangle, the hypotenuse is $13$. Since the area of the triangle is $30$, so our final altitude is $\frac{30(2)}{13}=\frac{60}{13}$. The sum of our altitudes is $\frac{60+156+65}{13}=\boxed{\textbf{(E)} \dfrac{281}{13}}$. Note that there is no need to calculate the final answer after we know that the third altitude has length $\frac{60}{13}$ since $E$ is the only choice with a denominator of $13$.

See Also

2015 AMC 10B (ProblemsAnswer KeyResources)
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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