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Difference between revisions of "2011 AMC 10A Problems/Problem 11"

(Created page with '==Problem 11== Square <math>EFGH</math> has one vertex on each side of square <math>ABCD</math>. Point <math>E</math> is on <math>AB</math> with <math>AE=7\cdot EB</math>. What…')
 
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<math>\text{(A)}\,\frac{49}{64}    \qquad\text{(B)}\,\frac{25}{32}    \qquad\text{(C)}\,\frac78 \qquad\text{(D)}\,\frac{5\sqrt{2}}{8}  \qquad\text{(E)}\,\frac{\sqrt{14}}{4} </math>
 
<math>\text{(A)}\,\frac{49}{64}    \qquad\text{(B)}\,\frac{25}{32}    \qquad\text{(C)}\,\frac78 \qquad\text{(D)}\,\frac{5\sqrt{2}}{8}  \qquad\text{(E)}\,\frac{\sqrt{14}}{4} </math>
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== Solution ==
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Let <math>8s</math> be the length of the sides of square <math>ABCD</math>, then the length of one of the sides of square <math>EFGH</math> is <math>\sqrt{(7s)^2+s^2}=\sqrt{50s^2}</math>, and hence the ratio in the areas is <math>\frac{\sqrt{50s^2}^2}{(8s)^2}=\frac{50}{64} = \boxed{\frac{25}{32} \ \mathbf{(B)}}</math>.

Revision as of 00:06, 16 February 2011

Problem 11

Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $AB$ with $AE=7\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$?

$\text{(A)}\,\frac{49}{64} \qquad\text{(B)}\,\frac{25}{32} \qquad\text{(C)}\,\frac78 \qquad\text{(D)}\,\frac{5\sqrt{2}}{8} \qquad\text{(E)}\,\frac{\sqrt{14}}{4}$

Solution

Let $8s$ be the length of the sides of square $ABCD$, then the length of one of the sides of square $EFGH$ is $\sqrt{(7s)^2+s^2}=\sqrt{50s^2}$, and hence the ratio in the areas is $\frac{\sqrt{50s^2}^2}{(8s)^2}=\frac{50}{64} = \boxed{\frac{25}{32} \ \mathbf{(B)}}$.

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