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

(Created page with "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</ma...")
 
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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>. We multiply base by height and divide by <math>2</math> to get <math>\boxed{30}</math> as our answer.
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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>. 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{281/13}</math>.

Revision as of 22:37, 3 March 2015

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 $30$, our final altitude has to be $30$ divided by the hypotenuse. By the Pythagorean Theorem, our hypotenuse is $13$, so the sum of our altitudes is $\boxed{281/13}$.

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