Difference between revisions of "2010 AMC 12A Problems/Problem 3"
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− | == Problem | + | == Problem == |
Rectangle <math>ABCD</math>, pictured below, shares <math>50\%</math> of its area with square <math>EFGH</math>. Square <math>EFGH</math> shares <math>20\%</math> of its area with rectangle <math>ABCD</math>. What is <math>\frac{AB}{AD}</math>? | Rectangle <math>ABCD</math>, pictured below, shares <math>50\%</math> of its area with square <math>EFGH</math>. Square <math>EFGH</math> shares <math>20\%</math> of its area with rectangle <math>ABCD</math>. What is <math>\frac{AB}{AD}</math>? | ||
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label("$H$",(0,0),SW); | label("$H$",(0,0),SW); | ||
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<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10</math> | <math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10</math> | ||
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== Solution == | == Solution == | ||
− | + | If we shift <math>A</math> to coincide with <math>E</math>, and add new horizontal lines to divide <math>EFGH</math> into five equal parts: | |
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<center><asy> | <center><asy> | ||
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This helps us to see that <math>AD=a/5</math> and <math>AB=2a</math>, where <math>a=EF</math>. | This helps us to see that <math>AD=a/5</math> and <math>AB=2a</math>, where <math>a=EF</math>. | ||
Hence <math>\dfrac{AB}{AD}=\dfrac{2a}{a/5}=10</math>. | Hence <math>\dfrac{AB}{AD}=\dfrac{2a}{a/5}=10</math>. | ||
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+ | == See also == | ||
+ | {{AMC12 box|year=2010|num-b=2|num-a=4|ab=A}} | ||
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+ | [[Category:Introductory Geometry Problems]] | ||
+ | {{MAA Notice}} |
Revision as of 18:05, 3 August 2020
Problem
Rectangle , pictured below, shares of its area with square . Square shares of its area with rectangle . What is ?
Solution
If we shift to coincide with , and add new horizontal lines to divide into five equal parts:
This helps us to see that and , where . Hence .
See also
2010 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 2 |
Followed by Problem 4 |
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 12 Problems and Solutions |
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