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Difference between revisions of "2018 AIME I Problems/Problem 4"

(→‎Solution 1)
(→‎Solution 1)
Line 10: Line 10:
 
dotfactor=3;
 
dotfactor=3;
  
−
pair B = (0,0), A = (6,8), C = (12,0);
+
pair B = (0,0), A = (6,8), C = (12,0), D = (2.154,2.872), E = (8.205, 6.153);
 
D(A--B);
 
D(A--B);
 
D(C--B);
 
D(C--B);
Line 19: Line 19:
 
label("$B$",B,SW);
 
label("$B$",B,SW);
 
label("$C$",C,SE);
 
label("$C$",C,SE);
−
 
+
label("$D$",D,NW);
 +
label("$E$",E,NE);
 
</asy>
 
</asy>
 
</center>
 
</center>

Revision as of 17:53, 7 March 2018

Problem 4

In $\triangle ABC, AB = AC = 10$ and $BC = 12$. Point $D$ lies strictly between $A$ and $B$ on $\overline{AB}$ and point $E$ lies strictly between $A$ and $C$ on $\overline{AC}$) so that $AD = DE = EC$. Then $AD$ can be expressed in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Solution 1

[asy] import cse5; unitsize(10mm); pathpen=black; dotfactor=3; pair B = (0,0), A = (6,8), C = (12,0), D = (2.154,2.872), E = (8.205, 6.153); D(A--B); D(C--B); D(A--C); dot(dotted); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NW); label("$E$",E,NE); [/asy]

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