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

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== Solution ==
 
== Solution ==
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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("<math>A</math>",A,dir(135));
 
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dot("<math>B</math>",B,dir(215));
 
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dot("<math>C</math>",C,dir(305));
 
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dot("<math>D</math>",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("<math>H</math>",H,dir(90));
 
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dot("<math>F</math>",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("<math>E</math>",E,dir(180));
 
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dot("<math>G</math>",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("<math>P</math>",P,dir(60));
 
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label("<math>w=296k</math>", intersectionpoint( A--P, E--H ));
 
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label("<math>x=275</math>", intersectionpoint( B--P, E--F ));
 
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label("<math>y=405k</math>", intersectionpoint( C--P, G--F ));
 
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label("<math>z=411k</math>", intersectionpoint( D--P, G--H ));[/asy]
 
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2014|n=I|num-b=12|num-a=14}}
 
{{AIME box|year=2014|n=I|num-b=12|num-a=14}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 20:56, 15 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 (Problems • Answer Key • Resources)
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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