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Difference between revisions of "2019 AIME II Problems/Problem 1"

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==Solution==
 
==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>

Revision as of 17:46, 22 March 2019

Problem

Two different points, $C$ and $D$, lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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]

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