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 ==
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Notice that <math>269+411=275+405</math>. This means <math>\overline{EG}</math> passes through the center of the square.
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Draw <math>\overline{IJ} \parallel \overline{HF}</math> with <math>I</math> on <math>\overline{AD}</math>, <math>J</math> on <math>\overline{BC}</math> such that <math>\overline{IJ}</math> and <math>\overline{EG}</math> intersects at the center of the square which I'll label as <math>O</math>.
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Let the area of the square be <math>1360a</math>. Then the area of <math>HPOI=71a</math> and the area of <math>FPOJ=65a</math>. This is because <math>\overline{HF}</math> is perpendicular to <math>\overline{EG}</math> (given in the problem), so <math>\overline{IJ}</math> is also perpendicular to <math>\overline{EG}</math>. These two orthogonal lines also pass through the center of the square, so they split it into 4 congruent quadrilaterals.
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Let the side length of the square be <math>d=\sqrt{1360a}</math>.
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Draw <math>\overline{OK}\parallel \overline{HI}</math> and intersects <math>\overline{HF}</math> at <math>K</math>. <math>OK=d\cdot\frac{[HFJI]}{[ABCD]}=\frac{d}{10}</math>.
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The area of <math>HKOI=\frac12\cdot HFJI=68a</math>, so the area of <math>POK=3a</math>.
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Let <math>\overline{PO}=h</math>. Then <math>KP=\frac{6a}{h}</math>
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Consider the area of <math>PFJO</math>.
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<cmath>\frac12(PF+OJ)(PO)=65a</cmath>
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<cmath>\left(17-\frac{3a}{h}\right)h=65a</cmath>
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<cmath>h=4a</cmath>
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Thus, <math>KP=1.5</math>.
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Solving <math>(4a)^2+1.5^2=\left(\frac{d}{10}\right)^2=13.6a</math>, we get <math>a=\frac58</math>.
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Therefore, the area of <math>ABCD=1360a=\boxed{850}</math>
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==Lazy Solution==
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<math>269+275+405+411=1360</math>, a multiple of <math>17</math>. In addition, <math>EG=FH=34</math>, which is <math>17\cdot 2</math>.
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Therefore, we suspect the square of the "hypotenuse" of a right triangle, corresponding to <math>EG</math> and <math>FH</math>  must be a multiple of <math>17</math>. All of these triples are primitive:
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<cmath>17=1^2+4^2</cmath>
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<cmath>34=3^2+5^2</cmath>
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<cmath>51=\emptyset</cmath>
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<cmath>68=\emptyset\text{ others}</cmath>
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<cmath>85=2^2+9^2=6^2+7^2</cmath>
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<cmath>102=\emptyset</cmath>
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<cmath>119=\emptyset \dots</cmath>
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The sides of the square can only equal the longer leg, or else the lines would have to extend outside of the square. Substituting <math>EG=FH=34</math>:
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<cmath>\sqrt{17}\rightarrow 34\implies 8\sqrt{17}\implies A=\textcolor{red}{1088}</cmath>
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<cmath>\sqrt{34}\rightarrow 34\implies 5\sqrt{34}\implies A=850</cmath>
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<cmath>\sqrt{85}\rightarrow 34\implies \{18\sqrt{85}/5,14\sqrt{85}/5\}\implies A=\textcolor{red}{1101.6,666.4}</cmath>
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Thus, <math>\boxed{850}</math> is the only valid answer.
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== See also ==
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{{AIME box|year=2014|n=I|num-b=12|num-a=14}}
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{{MAA Notice}}

Revision as of 19:38, 7 July 2020

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

Notice that $269+411=275+405$. This means $\overline{EG}$ passes through the center of the square.

Draw $\overline{IJ} \parallel \overline{HF}$ with $I$ on $\overline{AD}$, $J$ on $\overline{BC}$ such that $\overline{IJ}$ and $\overline{EG}$ intersects at the center of the square which I'll label as $O$.

Let the area of the square be $1360a$. Then the area of $HPOI=71a$ and the area of $FPOJ=65a$. This is because $\overline{HF}$ is perpendicular to $\overline{EG}$ (given in the problem), so $\overline{IJ}$ is also perpendicular to $\overline{EG}$. These two orthogonal lines also pass through the center of the square, so they split it into 4 congruent quadrilaterals.

Let the side length of the square be $d=\sqrt{1360a}$.

Draw $\overline{OK}\parallel \overline{HI}$ and intersects $\overline{HF}$ at $K$. $OK=d\cdot\frac{[HFJI]}{[ABCD]}=\frac{d}{10}$.

The area of $HKOI=\frac12\cdot HFJI=68a$, so the area of $POK=3a$.

Let $\overline{PO}=h$. Then $KP=\frac{6a}{h}$

Consider the area of $PFJO$. \[\frac12(PF+OJ)(PO)=65a\] \[\left(17-\frac{3a}{h}\right)h=65a\] \[h=4a\]

Thus, $KP=1.5$.

Solving $(4a)^2+1.5^2=\left(\frac{d}{10}\right)^2=13.6a$, we get $a=\frac58$.

Therefore, the area of $ABCD=1360a=\boxed{850}$

Lazy Solution

$269+275+405+411=1360$, a multiple of $17$. In addition, $EG=FH=34$, which is $17\cdot 2$. Therefore, we suspect the square of the "hypotenuse" of a right triangle, corresponding to $EG$ and $FH$ must be a multiple of $17$. All of these triples are primitive:

\[17=1^2+4^2\] \[34=3^2+5^2\] \[51=\emptyset\] \[68=\emptyset\text{ others}\] \[85=2^2+9^2=6^2+7^2\] \[102=\emptyset\] \[119=\emptyset \dots\]

The sides of the square can only equal the longer leg, or else the lines would have to extend outside of the square. Substituting $EG=FH=34$: \[\sqrt{17}\rightarrow 34\implies 8\sqrt{17}\implies A=\textcolor{red}{1088}\] \[\sqrt{34}\rightarrow 34\implies 5\sqrt{34}\implies A=850\] \[\sqrt{85}\rightarrow 34\implies \{18\sqrt{85}/5,14\sqrt{85}/5\}\implies A=\textcolor{red}{1101.6,666.4}\]

Thus, $\boxed{850}$ is the only valid answer.

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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