Difference between revisions of "2014 AIME I Problems/Problem 13"
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== Solution == | == 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("<math>A</math>",A,dir(135)); | ||
+ | dot("<math>B</math>",B,dir(215)); | ||
+ | dot("<math>C</math>",C,dir(305)); | ||
+ | dot("<math>D</math>",D,dir(45)); | ||
+ | pair H = ((2sqrt(850)-sqrt(306))/6,sqrt(850)); | ||
+ | pair F = ((2sqrt(850)+sqrt(306)+7)/6,0); | ||
+ | dot("<math>H</math>",H,dir(90)); | ||
+ | dot("<math>F</math>",F,dir(270)); | ||
+ | draw(H--F); | ||
+ | pair E = (0,(sqrt(850)-6)/2); | ||
+ | pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2); | ||
+ | dot("<math>E</math>",E,dir(180)); | ||
+ | dot("<math>G</math>",G,dir(0)); | ||
+ | draw(E--G); | ||
+ | pair P = extension(H,F,E,G); | ||
+ | dot("<math>P</math>",P,dir(60)); | ||
+ | label("<math>296k</math>", intersectionpoint( A--P, E--H )); | ||
+ | label("<math>275</math>", intersectionpoint( B--P, E--F )); | ||
+ | label("<math>405k</math>", intersectionpoint( C--P, G--F )); | ||
+ | label("<math>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 21:55, 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.