Difference between revisions of "1975 AHSME Problems/Problem 4"

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==Problem==
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If the side of one square is the diagonal of a second square, what is the ratio of the area of the first square to the area of the second?  
 
If the side of one square is the diagonal of a second square, what is the ratio of the area of the first square to the area of the second?  
  
<math>\textbf{(A)}\ 2 \qquad  
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<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ \sqrt2 \qquad \textbf{(C)}\ 1/2 \qquad \textbf{(D)}\ 2\sqrt2 \qquad \textbf{(E)}\ 4    </math>
\textbf{(B)}\ \sqrt2 \qquad  
 
\textbf{(C)}\ 1/2 \qquad  
 
\textbf{(D)}\ 2\sqrt2 \qquad  
 
\textbf{(E)}\ 4    </math>
 
  
  
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Denote the side of one square as <math>s</math>. Then the diagonal of the second square is <math>s</math>, so the side of the second square is <math>\dfrac{s\sqrt{2}}{2}</math>. The area of the second square is <math>\dfrac{1}{2}s^2</math>, so the ratio of the areas is <math>\dfrac{s^2}{\dfrac{1}{2}s^2} = \boxed{\textbf{(A) } 2}</math>.
 
Denote the side of one square as <math>s</math>. Then the diagonal of the second square is <math>s</math>, so the side of the second square is <math>\dfrac{s\sqrt{2}}{2}</math>. The area of the second square is <math>\dfrac{1}{2}s^2</math>, so the ratio of the areas is <math>\dfrac{s^2}{\dfrac{1}{2}s^2} = \boxed{\textbf{(A) } 2}</math>.
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==See Also==
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{{AHSME box|year=1975|num-b=3|num-a=5}}
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{{MAA Notice}}

Latest revision as of 15:51, 19 January 2021

Problem

If the side of one square is the diagonal of a second square, what is the ratio of the area of the first square to the area of the second?

$\textbf{(A)}\ 2 \qquad  \textbf{(B)}\ \sqrt2 \qquad  \textbf{(C)}\ 1/2 \qquad  \textbf{(D)}\ 2\sqrt2 \qquad \textbf{(E)}\ 4$


Solution

Solution by e_power_pi_times_i


Denote the side of one square as $s$. Then the diagonal of the second square is $s$, so the side of the second square is $\dfrac{s\sqrt{2}}{2}$. The area of the second square is $\dfrac{1}{2}s^2$, so the ratio of the areas is $\dfrac{s^2}{\dfrac{1}{2}s^2} = \boxed{\textbf{(A) } 2}$.

See Also

1975 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
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
All AHSME Problems and Solutions

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