Difference between revisions of "2019 AIME II Problems/Problem 1"
Brendanb4321 (talk | contribs) (→Solution) |
Brendanb4321 (talk | contribs) (→Solution) |
||
| Line 24: | Line 24: | ||
label("$8$",(D+(-6,0))/2,dir(180)); | label("$8$",(D+(-6,0))/2,dir(180)); | ||
label("$6$",(A+(-6,0))/2,dir(-90)); | label("$6$",(A+(-6,0))/2,dir(-90)); | ||
| − | |||
</asy> | </asy> | ||
| + | - Diagram by Brendanb4321 | ||
Revision as of 16:46, 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
