Difference between revisions of "2019 AIME II Problems/Problem 1"
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Revision as of 16:55, 22 March 2019
Problem
Two different points, 
and 
, lie on the same side of line 
so that 
and 
are congruent with 
, 
, and 
. The intersection of these two triangular regions has area 
, where 
and 
are relatively prime positive integers. Find 
.
Solution
![[asy] unitsize(10); pair A = (0,0); pair B = (9,0); pair C = (15,8); pair D = (-6,8); draw(A--B--C--cycle); draw(B--D--A); label("$A$",A,dir(-120)); label("$B$",B,dir(-60)); label("$C$",C,dir(60)); label("$D$",D,dir(120)); label("$9$",(A+B)/2,dir(-90)); label("$10$",(D+A)/2,dir(-150)); label("$10$",(C+B)/2,dir(-30)); label("$17$",(D+B)/2,dir(60)); label("$17$",(A+C)/2,dir(120)); draw(D--(-6,0)--A,dotted); label("$8$",(D+(-6,0))/2,dir(180)); label("$6$",(A+(-6,0))/2,dir(-90)); [/asy]](http://latex.artofproblemsolving.com/1/9/9/199ac170a50a6642d8a6e0afed168e52fce19c08.png)
- Diagram by Brendanb4321
See Also
| 2019 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by First Problem |
Followed by Problem 2 | |
| 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. 
