Difference between revisions of "2014 AIME I Problems/Problem 13"
| Line 1: | Line 1: | ||
| − | == Problem 13 == | + | ==Problem 13== |
| + | 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>. | ||
| + | |||
| + | <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 == | == Solution == | ||
Revision as of 19:59, 14 March 2014
Problem 13
On square 
, points 
, and 
lie on sides 
and 
respectively, so that 
and 
. Segments 
and 
intersect at a point 
, and the areas of the quadrilaterals 
and 
are in the ratio 
Find the area of square .
![[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]](http://latex.artofproblemsolving.com/e/1/1/e1175984ad8a8e7468972e727cc95e27117b7d92.png)
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
