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Difference between revisions of "2011 AMC 10A Problems/Problem 11"
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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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[[Category: Introductory Geometry Problems]]
  
 
== Solution ==
 
== Solution ==
  
 
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>.
 
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>.
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== Video Solution by OmegaLearn ==
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https://youtu.be/GrCtzL0S-Uo?t=292
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~ pi_is_3.14
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== See Also ==
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{{AMC10 box|year=2011|ab=A|num-b=10|num-a=12}}
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{{MAA Notice}}

Latest revision as of 02:42, 23 January 2023

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 $\frac{\sqrt{50s^2}^2}{(8s)^2}=\frac{50}{64} = \boxed{\frac{25}{32} \ \mathbf{(B)}}$.

Video Solution by OmegaLearn

https://youtu.be/GrCtzL0S-Uo?t=292

~ pi_is_3.14

See Also

2011 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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 10 Problems and Solutions

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