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Difference between revisions of "2005 AMC 10A Problems/Problem 8"

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<math>HE=6</math> So, the area of the square is <math>6^2=\boxed{36}</math>.
 
<math>HE=6</math> So, the area of the square is <math>6^2=\boxed{36}</math>.
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==See Also==
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[[Category:Introductory Geometry Problems]]
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[[Category:Area Problems]]

Revision as of 20:35, 11 April 2013

Problem

In the figure, the length of side $AB$ of square $ABCD$ is $\sqrt{50}$ and $BE$=1. What is the area of the inner square $EFGH$?

AMC102005Aq.png

$\textbf{(A)}\ 25\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 42$

Solution

(C) We see that side $BE$, which we know is 1, is also the shorter leg of one of the four right triangles (which are congruent, I'll not prove this). So, $AH = 1$. Then $HB = HE + BE = HE + 1$, and $HE$ is one of the sides of the square whose area we want to find. So:

$1^2 + (HE+1)^2=\sqrt{50}^2$

$1 + (HE+1)^2=50$

$(HE+1)^2=49$

$HE+1=7$

$HE=6$ So, the area of the square is $6^2=\boxed{36}$.

See Also

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