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

Revision as of 18:07, 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

$[asy] import cse5; unitsize(10mm); pathpen=black; dotfactor=3; pair B = (0,0), A = (6,8), C = (12,0), D = (2.154,2.872), E = (8.153, 5.128), F=(7.68,5.76), G=(7.077,6.564), H=(5.157,5.124), I=(4.5,6); pair[] dotted = {A,B,C,D,E,F,G}; D(A--B); D(C--B); D(A--C); D(D--E); pathpen=dashed; D(B--F); D(D--G); dot(dotted); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NW); label("$E$",E,NE); label("$F$",F,NE); label("$G$",G,NE); label("$x$",H,NE); label("$x$",I,NE); [/asy]$

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

Revision as of 18:07, 7 March 2018

Problem 4

Solution 1

@@ Line 10: / Line 10: @@
 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), G=(7.077,6.564);
+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), G=(7.077,6.564), H=(5.157,5.124), I=(4.5,6);
 pair[] dotted = {A,B,C,D,E,F,G};
@@ Line 29: / Line 29: @@
 label("$F$",F,NE);
 label("$G$",G,NE);
+label("$x$",H,NE);
+label("$x$",I,NE);
 </asy>
 </center>