Difference between revisions of "2015 AIME I Problems/Problem 7"
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==Problem== | ==Problem== | ||
7. In the diagram below, <math>ABCD</math> is a square. Point <math>E</math> is the midpoint of <math>\overline{AD}</math>. Points <math>F</math> and <math>G</math> lie on <math>\overline{CE}</math>, and <math>H</math> and <math>J</math> lie on <math>\overline{AB}</math> and <math>\overline{BC}</math>, respectively, so that <math>FGHJ</math> is a square. Points <math>K</math> and <math>L</math> lie on <math>\overline{GH}</math>, and <math>M</math> and <math>N</math> lie on <math>\overline{AD}</math> and <math>\overline{AB}</math>, respectively, so that <math>KLMN</math> is a square. The area of <math>KLMN</math> is 99. Find the area of <math>FGHJ</math>. | 7. In the diagram below, <math>ABCD</math> is a square. Point <math>E</math> is the midpoint of <math>\overline{AD}</math>. Points <math>F</math> and <math>G</math> lie on <math>\overline{CE}</math>, and <math>H</math> and <math>J</math> lie on <math>\overline{AB}</math> and <math>\overline{BC}</math>, respectively, so that <math>FGHJ</math> is a square. Points <math>K</math> and <math>L</math> lie on <math>\overline{GH}</math>, and <math>M</math> and <math>N</math> lie on <math>\overline{AD}</math> and <math>\overline{AB}</math>, respectively, so that <math>KLMN</math> is a square. The area of <math>KLMN</math> is 99. Find the area of <math>FGHJ</math>. | ||
+ | |||
+ | <asy> | ||
+ | pair A,B,C,D,E,F,G,H,J,K,L,M,N; | ||
+ | B=(0,0); | ||
+ | real m=7*sqrt(55)/5; | ||
+ | J=(m,0); | ||
+ | C=(7*m/2,0); | ||
+ | A=(0,7*m/2); | ||
+ | D=(7*m/2,7*m/2); | ||
+ | E=(A+D)/2; | ||
+ | H=(0,2m); | ||
+ | N=(0,2m+3*sqrt(55)/2); | ||
+ | G=foot(H,E,C); | ||
+ | F=foot(J,E,C); | ||
+ | draw(A--B--C--D--cycle); | ||
+ | draw(C--E); | ||
+ | draw(G--H--J--F); | ||
+ | pair X=foot(N,E,C); | ||
+ | M=extension(N,X,A,D); | ||
+ | K=foot(N,H,G); | ||
+ | L=foot(M,H,G); | ||
+ | draw(K--N--M--L); | ||
+ | label("$A$",A,NW); | ||
+ | label("$B$",B,SW); | ||
+ | label("$C$",C,SE); | ||
+ | label("$D$",D,NE); | ||
+ | label("$E$",E,dir(90)); | ||
+ | label("$F$",F,NE); | ||
+ | label("$G$",G,NE); | ||
+ | label("$H$",H,W); | ||
+ | label("$J$",J,S); | ||
+ | label("$K$",K,SE); | ||
+ | label("$L$",L,SE); | ||
+ | label("$M$",M,dir(90)); | ||
+ | label("$N$",N,dir(180)); </asy> | ||
+ | |||
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Revision as of 18:54, 20 March 2015
Problem
7. In the diagram below, is a square. Point is the midpoint of . Points and lie on , and and lie on and , respectively, so that is a square. Points and lie on , and and lie on and , respectively, so that is a square. The area of is 99. Find the area of .
Solution
We begin by denoting the length , giving us and . Since angles and are complimentary, we have that (and similarly the rest of the triangles are triangles). We let the sidelength of be , giving us:
and .
Since ,
,
Solving for in terms of yields .
We now use the given that , implying that . We also draw the perpendicular from E to ML and label the point of intersection P.
This gives that and
Since = , we get
So our final answer is
See also
2015 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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.