Difference between revisions of "2017 AIME II Problems/Problem 10"
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==Solution== | ==Solution== | ||
| + | <asy> | ||
| + | pair A,B,C,D,M,n,O,P; | ||
| + | A=(0,42);B=(84,42);C=(84,0);D=(0,0);M=(0,21);n=(28,42);O=(12,18);P=(32,13); | ||
| + | draw(A--B--C--D--cycle); | ||
| + | draw(C--M); | ||
| + | draw(D--n); | ||
| + | draw(B--P); | ||
| + | draw(D--P); | ||
| + | label("$A$",A,NW); | ||
| + | label("$B$",B,NE); | ||
| + | label("$C$",C,SE); | ||
| + | label("$D$",D,SW); | ||
| + | label("$M$",M,W); | ||
| + | label("$N$",n,N); | ||
| + | label("$O$",O,(-0.5,1)); | ||
| + | label("$P$",P,N); | ||
| + | dot(A); | ||
| + | dot(B); | ||
| + | dot(C); | ||
| + | dot(D); | ||
| + | dot(M); | ||
| + | dot(n); | ||
| + | dot(O); | ||
| + | dot(P); | ||
| + | </asy> | ||
<math>\boxed{546}</math> | <math>\boxed{546}</math> | ||
Revision as of 15:45, 23 March 2017
Problem
Rectangle 
has side lengths 
and 
. Point 
is the midpoint of 
, point 
is the trisection point of 
closer to 
, and point 
is the intersection of 
and 
. Point 
lies on the quadrilateral 
, and 
bisects the area of . Find the area of 
.
Solution
![[asy] pair A,B,C,D,M,n,O,P; A=(0,42);B=(84,42);C=(84,0);D=(0,0);M=(0,21);n=(28,42);O=(12,18);P=(32,13); draw(A--B--C--D--cycle); draw(C--M); draw(D--n); draw(B--P); draw(D--P); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$M$",M,W); label("$N$",n,N); label("$O$",O,(-0.5,1)); label("$P$",P,N); dot(A); dot(B); dot(C); dot(D); dot(M); dot(n); dot(O); dot(P); [/asy]](http://latex.artofproblemsolving.com/1/7/5/175bce0999c0083cbcc8790bc86ad0da52c77ee6.png)
See Also
| 2017 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 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. 
