MathDB
Minimizing Side Length

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January 11, 2009
Pythagorean Theoremgeometry

Problem Statement

Consider all triangles ABC ABC satisfying the following conditions: AB \equal{} AC, D D is a point on AC \overline{AC} for which BDAC \overline{BD} \perp \overline{AC}, AD AD and CD CD are integers, and BD^2 \equal{} 57. Among all such triangles, the smallest possible value of AC AC is <spanclass=latexbold>(A)</span> 9<spanclass=latexbold>(B)</span> 10<spanclass=latexbold>(C)</span> 11<spanclass=latexbold>(D)</span> 12<spanclass=latexbold>(E)</span> 13 <span class='latex-bold'>(A)</span>\ 9 \qquad <span class='latex-bold'>(B)</span>\ 10 \qquad <span class='latex-bold'>(C)</span>\ 11 \qquad <span class='latex-bold'>(D)</span>\ 12 \qquad <span class='latex-bold'>(E)</span>\ 13 [asy]defaultpen(linewidth(.8pt)); dotfactor=4;
pair B = (0,0); pair C = (5,0); pair A = (2.5,7.5); pair D = foot(B,A,C); dot(A);dot(B);dot(C);dot(D); label("AA", A, N);label("BB", B, SW);label("CC", C, SE);label("DD", D, NE); draw(A--B--C--cycle);draw(B--D);[/asy]