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

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== Problem ==
 
== Problem ==
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In the figure, the length of side <math>AB</math> of square <math>ABCD</math> is <math>\sqrt{50}</math> and <math>BE</math>=1. What is the area of the inner square <math>EFGH</math>?
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In the figure, the length of side <math>AB</math> of square <math>ABCD</math> is <math>\sqrt{50}</math> and <math>BE=1</math>. What is the area of the inner square <math>EFGH</math>?
  
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[asy]
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[[File:AMC102005Aq.png]]
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pair D = (0,0); pair C = (1,0); pair B = (1,1); pair A = (0,1); pair H = (0.25,0.6); pair E = foot(C,H,B); pair F = foot(D,C,E); pair G = foot(A,D,F); pair H2 = foot(B,A,G); draw(A--B--C--D--cycle,linewidth(1)); draw(B--H2,linewidth(1)); draw(C--E,linewidth(1)); draw(D--F,linewidth(1)); draw(A--G,linewidth(1)); label("<math>A</math>",A,NW);label("<math>B</math>",B,NE);label("<math>C</math>",C,SE);label("<math>D</math>",D,SW); label("<math>H</math>",H2,SW);label("<math>G</math>",G,SE);label("<math>F</math>",F,NE);label("<math>E</math>",E,NW);[/asy]
 
  
 
<math> \textbf{(A)}\ 25\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 42 </math>
 
<math> \textbf{(A)}\ 25\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 42 </math>
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==Solution==
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We see that side <math>BE</math>, which we know is <math>1</math>, is also the shorter leg of one of the four right triangles (which are congruent, I won’t prove this). So, <math>AH = 1</math>. Then <math>HB = HE + BE = HE + 1</math>, and <math>HE</math> is one of the sides of the square whose area we want to find. So:
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<cmath>1^2 + (HE+1)^2=\sqrt{50}^2</cmath>
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<cmath>1 + (HE+1)^2=50</cmath>
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<cmath>(HE+1)^2=49</cmath>
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<cmath>HE+1=7</cmath>
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<cmath>HE=6</cmath>
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So, the area of the square is <math>6^2=\boxed{\textbf{(C) }36}</math>
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==See also==
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{{AMC10 box|year=2005|ab=A|num-b=7|num-a=9}}
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{{MAA Notice}}

Latest revision as of 10:58, 13 December 2021

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

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 won’t 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 $6^2=\boxed{\textbf{(C) }36}$

See also

2005 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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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