Difference between revisions of "2014 AIME I Problems/Problem 13"
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pair P = extension(H,F,E,G); | pair P = extension(H,F,E,G); | ||
dot("<math>P</math>",P,dir(60)); | dot("<math>P</math>",P,dir(60)); | ||
− | label("<math>296k</math>", intersectionpoint( A--P, E--H )); | + | label("<math>w=296k</math>", intersectionpoint( A--P, E--H )); |
− | label("<math>275</math>", intersectionpoint( B--P, E--F )); | + | label("<math>x=275</math>", intersectionpoint( B--P, E--F )); |
− | label("<math>405k</math>", intersectionpoint( C--P, G--F )); | + | label("<math>y=405k</math>", intersectionpoint( C--P, G--F )); |
− | label("<math>411k</math>", intersectionpoint( D--P, G--H ));[/asy] | + | 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 , 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 .
Solution
[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,dir(135)); dot("",B,dir(215)); dot("",C,dir(305)); dot("",D,dir(45)); pair H = ((2sqrt(850)-sqrt(306))/6,sqrt(850)); pair F = ((2sqrt(850)+sqrt(306)+7)/6,0); dot("",H,dir(90)); dot("",F,dir(270)); draw(H--F); pair E = (0,(sqrt(850)-6)/2); pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2); dot("",E,dir(180)); dot("",G,dir(0)); draw(E--G); pair P = extension(H,F,E,G); dot("",P,dir(60)); label("", intersectionpoint( A--P, E--H )); label("", intersectionpoint( B--P, E--F )); label("", intersectionpoint( C--P, G--F )); label("", intersectionpoint( D--P, G--H ));[/asy]
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.