Difference between revisions of "2013 AMC 10A Problems/Problem 15"

(Solution 2 (actually solving))
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<math> \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 18 </math>
 
<math> \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 18 </math>
  
==Solution 1 (cheap way)==
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==Solution 1 (Process of Elimination)==
  
 
The shortest side length has the longest altitude perpendicular to it. The average of the two altitudes given will be between the lengths of the two altitudes,
 
The shortest side length has the longest altitude perpendicular to it. The average of the two altitudes given will be between the lengths of the two altitudes,
therefore the length of the side perpendicular to that altitude will be between <math>10</math> and <math>15</math>. The only answer choice that meets this requirement is 12, <math>\textbf{(D)}</math>.
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therefore the length of the side perpendicular to that altitude will be between <math>10</math> and <math>15</math>. The only answer choice that meets this requirement is <math>\boxed{\textbf{(D) }12}</math>.
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'''
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[[Category: Introductory Geometry Problems]]
  
==Solution 2 (actually solving)==
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==Solution 2==
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'''
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Let the height to the side of length <math>15</math> be <math>h_{1}</math>, the height to the side of length 10 be <math>h_{2}</math>, the area be <math>A</math>, and the height to the unknown side be <math>h_{3}</math>.
  
Let the height to the side of length 15 be <math>h_{1}</math>, the height to the side of length 10 be <math>h_{2}</math>, the area be <math>A</math>, and the height to the unknown side be <math>h_{3}</math>.
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Because the area of a triangle is <math>\frac{bh}{2}</math>, we get that <math>15(h_{1}) = 2A</math> and <math>10(h_{2}) = 2A</math>, so, setting them equal, <math>h_{2} = \frac{3h_{1}}{2}</math>.  From the problem, we know that <math>2h_{3} = h_{1} + h_{2}</math>.  Substituting, we get that
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<cmath>h_{3} = 1.25h_{1}.</cmath> 
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Thus, the side length is going to be <math>\frac{2A}{1.25h_{1}} = \frac{15}{\frac{5}{4}} = \boxed{\textbf{(D) }12}</math>.
  
Because the area of a triangle is <math>\frac{bh}{2}</math>, we get that <math>15(h_{1}) = 2A</math> and <math>10(h_{2}) = 2A</math>, so, setting them equal, <math>h_{2} = \frac{3h_{1}}{2}</math>. Now, we know that <math>2h_{3} = h_{1} + h_{2}</math>.  Substituting, we get that
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==Video Solution==
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https://youtu.be/27TpizTnMeM
  
<math>h_{3} = 1.25h_{1}</math>. 
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~savannahsolver
 
 
Thus, the side length is going to be <math>\frac{2A}{1.25h_{1}} = \frac{15}{\frac{5}{4}} = 12</math>, <math>/textbf{(D)}</math>.
 
  
 
==See Also==
 
==See Also==
  
 
{{AMC10 box|year=2013|ab=A|num-b=14|num-a=16}}
 
{{AMC10 box|year=2013|ab=A|num-b=14|num-a=16}}
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{{MAA Notice}}

Revision as of 15:03, 25 December 2020

Problem

Two sides of a triangle have lengths $10$ and $15$. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side?


$\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 18$

Solution 1 (Process of Elimination)

The shortest side length has the longest altitude perpendicular to it. The average of the two altitudes given will be between the lengths of the two altitudes, therefore the length of the side perpendicular to that altitude will be between $10$ and $15$. The only answer choice that meets this requirement is $\boxed{\textbf{(D) }12}$.

Solution 2

Let the height to the side of length $15$ be $h_{1}$, the height to the side of length 10 be $h_{2}$, the area be $A$, and the height to the unknown side be $h_{3}$.

Because the area of a triangle is $\frac{bh}{2}$, we get that $15(h_{1}) = 2A$ and $10(h_{2}) = 2A$, so, setting them equal, $h_{2} = \frac{3h_{1}}{2}$. From the problem, we know that $2h_{3} = h_{1} + h_{2}$. Substituting, we get that \[h_{3} = 1.25h_{1}.\] Thus, the side length is going to be $\frac{2A}{1.25h_{1}} = \frac{15}{\frac{5}{4}} = \boxed{\textbf{(D) }12}$.

Video Solution

https://youtu.be/27TpizTnMeM

~savannahsolver

See Also

2013 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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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