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Difference between revisions of "2012 AMC 10B Problems/Problem 19"

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The easiest way to find the area would be to find the area of <math>ABCD</math> and subtract the areas of <math>ABG</math> and <math>CDF.</math> You can easily get the area of <math>ABG</math> because you know <math>AB=6</math> and <math>AG=15</math>, so <math>ABG</math>'s area is <math>15\cdot 6/2=45</math>. However, for triangle <math>CDF,</math> you don't know <math>CF.</math> However, you can note that triangle <math>BEF</math> is similar to triangle <math>CDF</math> through AA. You see that <math>BE/DC=1/3.</math> So, You can do <math>BF+3BF=30</math> for <math>BF=15/2,</math> and <math>CF=3BF=3(15/2)=45/2.</math> Now, you can find the area of <math>CDF,</math> which is <math>135/2.</math> Now, you do <math>180-225/2,</math> which turns out to be <math>135/2,</math> which makes the answer (C).
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The easiest way to find the area would be to find the area of <math>ABCD</math> and subtract the areas of <math>ABG</math> and <math>CDF.</math> You can easily get the area of <math>ABG</math> because you know <math>AB=6</math> and <math>AG=15</math>, so <math>ABG</math>'s area is <math>15\cdot 6/2=45</math>. However, for triangle <math>CDF,</math> you don't know <math>CF.</math> However, you can note that triangle <math>BEF</math> is similar to triangle <math>CDF</math> through AA. You see that <math>BE/DC=1/3.</math> So, You can do <math>BF+3BF=30</math> for <math>BF=15/2,</math> and <math>CF=3BF=3(15/2)=45/2.</math> Now, you can find the area of <math>CDF,</math> which is <math>135/2.</math> Now, you do <math>[ABCD]-[ABG]-[CDF]=180-45-135/2=135-135/2=135/2,</math>
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which makes the answer (C).

Revision as of 15:50, 15 January 2013

The easiest way to find the area would be to find the area of $ABCD$ and subtract the areas of $ABG$ and $CDF.$ You can easily get the area of $ABG$ because you know $AB=6$ and $AG=15$, so $ABG$'s area is $15\cdot 6/2=45$. However, for triangle $CDF,$ you don't know $CF.$ However, you can note that triangle $BEF$ is similar to triangle $CDF$ through AA. You see that $BE/DC=1/3.$ So, You can do $BF+3BF=30$ for $BF=15/2,$ and $CF=3BF=3(15/2)=45/2.$ Now, you can find the area of $CDF,$ which is $135/2.$ Now, you do $[ABCD]-[ABG]-[CDF]=180-45-135/2=135-135/2=135/2,$ which makes the answer (C).

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