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Difference between revisions of "1955 AHSME Problems/Problem 21"

(Created page with "== Problem 21== Represent the hypotenuse of a right triangle by <math>c</math> and the area by <math>A</math>. The altitude on the hypotenuse is: <math> \textbf{(A)}\ \frac...")
 
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<math> \textbf{(A)}\ \frac{A}{c}\qquad\textbf{(B)}\ \frac{2A}{c}\qquad\textbf{(C)}\ \frac{A}{2c}\qquad\textbf{(D)}\ \frac{A^2}{c}\qquad\textbf{(E)}\ \frac{A}{c^2} </math>
 
<math> \textbf{(A)}\ \frac{A}{c}\qquad\textbf{(B)}\ \frac{2A}{c}\qquad\textbf{(C)}\ \frac{A}{2c}\qquad\textbf{(D)}\ \frac{A^2}{c}\qquad\textbf{(E)}\ \frac{A}{c^2} </math>
 
==Solution==
 
==Solution==
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Link to Overleaf: [[https://www.overleaf.com/read/rmkbjhcyynbv|https://www.overleaf.com/read/rmkbjhcyynbv]]
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<asy>
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draw((0,0) -- (9/5,12/5) -- (5,0) -- cycle);
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draw((9/5,12/5) -- (9/5,0));
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</asy>
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Given that the area of the triangle is <math>A</math>, and the formula for the area of a triangle is <math>\frac{bh}{2}=A</math>, we can replace <math>b</math> (the base) and <math>h</math> (the height) with <math>c</math> (the hypotenuse) and <math>a</math> (the altitude), we can rearrrange as follows:
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<cmath>\frac{c*a}{2}=A</cmath>
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<cmath>c*a=2A</cmath>
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<cmath>a=\textbf{(B)} \frac{2A}{c}</cmath>
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==See Also==
 
==See Also==
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In order to return to the problem set, click [[1955 AHSME Problems|here]]
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{{AHSME 50p box|year=1955|num-b=20|num-a=22}}
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{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 21:14, 15 August 2020

Problem 21

Represent the hypotenuse of a right triangle by $c$ and the area by $A$. The altitude on the hypotenuse is:

$\textbf{(A)}\ \frac{A}{c}\qquad\textbf{(B)}\ \frac{2A}{c}\qquad\textbf{(C)}\ \frac{A}{2c}\qquad\textbf{(D)}\ \frac{A^2}{c}\qquad\textbf{(E)}\ \frac{A}{c^2}$

Solution

[asy] draw((0,0) -- (9/5,12/5) -- (5,0) -- cycle); draw((9/5,12/5) -- (9/5,0)); [/asy] Given that the area of the triangle is $A$, and the formula for the area of a triangle is $\frac{bh}{2}=A$, we can replace $b$ (the base) and $h$ (the height) with $c$ (the hypotenuse) and $a$ (the altitude), we can rearrrange as follows: \[\frac{c*a}{2}=A\] \[c*a=2A\] \[a=\textbf{(B)} \frac{2A}{c}\]

See Also

1955 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50
All AHSME Problems and Solutions


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