2018 AIME I Problems/Problem 4

Revision as of 18:59, 7 March 2018 by Bluebacon008 (talk | contribs) (Solution 1)

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 = (8.153, 5.128), F=(7.68,5.76); pair[] dotted = {A,B,C,D,E,F};  D(A--B); D(C--B); D(A--C); D(D--E); pathpen=dashed; D(B--F);  dot(dotted); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NW); label("$E$",E,NE); [/asy]