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

Revision as of 22:50, 23 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

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

Draw $\overline{IJ} \parallel \overline{HF}$ with $I$ on $\overline{AD}$ , $J$ on $\overline{EC}$ such that $\overline{IJ}$ and $\overline{EG}$ intersects at the centre of the square $O$ .

Let the area of the square be $1360a$ . Then the area of $HPOI=71a$ and the area of $FPOJ=65a$ .

Let the side side length 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 $PFJO$ . $\frac12(PF+OJ)(PO)=65a$ $(17-\frac{3a}{h})h=65a$ $h=4a$

Thus, $KP=1.5$ .

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

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

@@ Line 32: / Line 32: @@
 == Solution ==
+Notice that <math>269+411=275+405</math>. This means <math>\overline{EG}</math> passes through the centre of the square.
+Draw <math>\overline{IJ} \parallel \overline{HF}</math> with <math>I</math> on <math>\overline{AD}</math>, <math>J</math> on <math>\overline{EC}</math> such that <math>\overline{IJ}</math> and <math>\overline{EG}</math> intersects at the centre of the square <math>O</math>.
+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>.
+Let the side side length be <math>d=\sqrt{1360a}</math>.
+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>.
+The area of <math>HKOI=\frac12\cdot HFJI=68a</math>, so the area of <math>POK=3a</math>.
+Let <math>\overline{PO}=h</math>. Then <math>KP=\frac{6a}{h}</math>
+Consider the area of <math>PFJO</math>.
+<cmath>\frac12(PF+OJ)(PO)=65a</cmath>
+<cmath>(17-\frac{3a}{h})h=65a</cmath>
+<cmath>h=4a</cmath>
+Thus, <math>KP=1.5</math>.
+Solving <math>(4a)^2+1.5^2=(\frac{d}{10})^2=13.6a</math>, we get <math>a=\frac58</math>.
+Therefore, the area of <math>ABCD=1360a=\boxed{850}</math>
 == See also ==
 {{AIME box|year=2014|n=I|num-b=12|num-a=14}}
 {{MAA Notice}}

2014 AIME I (Problems • Answer Key • Resources)
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Difference between revisions of "2014 AIME I Problems/Problem 13"

Revision as of 22:50, 23 March 2014

Problem 13

Solution

See also