Difference between revisions of "2014 AIME I Problems/Problem 13"

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== Problem 13 ==
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==Problem 13==
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On square <math>ABCD</math>, points <math>E,F,G</math>, and <math>H</math> lie on sides <math>\overline{AB},\overline{BC},\overline{CD},</math> and <math>\overline{DA},</math> respectively, so that <math>\overline{EG} \perp \overline{FH}</math> and <math>EG=FH = 34</math>. Segments <math>\overline{EG}</math> and <math>\overline{FH}</math> intersect at a point <math>P</math>, and the areas of the quadrilaterals <math>AEPH, BFPE, CGPF,</math> and <math>DHPG</math> are in the ratio <math>269:275:405:411.</math> Find the area of square <math>ABCD</math>.
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<asy>
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pair A = (0,sqrt(850));
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pair B = (0,0);
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pair C = (sqrt(850),0);
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pair D = (sqrt(850),sqrt(850));
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draw(A--B--C--D--cycle);
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dotfactor = 3;
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dot("$A$",A,dir(135));
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dot("$B$",B,dir(215));
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dot("$C$",C,dir(305));
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dot("$D$",D,dir(45));
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pair H = ((2sqrt(850)-sqrt(306))/6,sqrt(850));
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pair F = ((2sqrt(850)+sqrt(306)+7)/6,0);
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dot("$H$",H,dir(90));
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dot("$F$",F,dir(270));
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draw(H--F);
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pair E = (0,(sqrt(850)-6)/2);
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pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2);
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dot("$E$",E,dir(180));
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dot("$G$",G,dir(0));
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draw(E--G);
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pair P = extension(H,F,E,G);
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dot("$P$",P,dir(60));
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label("$w$", intersectionpoint( A--P, E--H ));
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label("$x$", intersectionpoint( B--P, E--F ));
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label("$y$", intersectionpoint( C--P, G--F ));
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label("$z$", intersectionpoint( D--P, G--H ));</asy>
  
 
== Solution ==
 
== Solution ==

Revision as of 18:59, 14 March 2014

Problem 13

On square $ABCD$, points $E,F,G$, and $H$ lie on sides $\overline{AB},\overline{BC},\overline{CD},$ and $\overline{DA},$ respectively, so that $\overline{EG} \perp \overline{FH}$ and $EG=FH = 34$. Segments $\overline{EG}$ and $\overline{FH}$ intersect at a point $P$, and the areas of the quadrilaterals $AEPH, BFPE, CGPF,$ and $DHPG$ are in the ratio $269:275:405:411.$ Find the area of square $ABCD$.

[asy] pair A = (0,sqrt(850)); pair B = (0,0); pair C = (sqrt(850),0); pair D = (sqrt(850),sqrt(850)); draw(A--B--C--D--cycle); dotfactor = 3; dot("$A$",A,dir(135)); dot("$B$",B,dir(215)); dot("$C$",C,dir(305)); dot("$D$",D,dir(45)); pair H = ((2sqrt(850)-sqrt(306))/6,sqrt(850)); pair F = ((2sqrt(850)+sqrt(306)+7)/6,0); dot("$H$",H,dir(90)); dot("$F$",F,dir(270)); draw(H--F); pair E = (0,(sqrt(850)-6)/2); pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2); dot("$E$",E,dir(180)); dot("$G$",G,dir(0)); draw(E--G); pair P = extension(H,F,E,G); dot("$P$",P,dir(60)); label("$w$", intersectionpoint( A--P, E--H )); label("$x$", intersectionpoint( B--P, E--F )); label("$y$", intersectionpoint( C--P, G--F )); label("$z$", intersectionpoint( D--P, G--H ));[/asy]

Solution

See also

2014 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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