2014 AIME I Problems/Problem 13
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 |
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