2018 AIME I Problems/Problem 4

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

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);
pair[] dotted = {A,B,C,D,E};

D(A--B);
D(C--B);
D(A--C);
pathpen=dashed;
D(D--E)

dot(dotted);
label("$A$",A,N);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,NW);
label("$E$",E,NE);
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2018 AIME I Problems/Problem 4

Problem 4

Solution 1