Problem

1999 AHSME Problem 19

Consider all triangles $ABC$ satisfying in the following conditions: $AB = AC$ , $D$ is a point on $\overline{AC}$ for which $\overline{BD} \perp \overline{AC}$ , $AC$ and $CD$ are integers, and $BD^{2} = 57$ . Among all such triangles, the smallest possible value of $AC$ is

[asy] pair A,B,C,D; A=(5,12); B=origin; C=(10,0); D=(8.52071005917,3.55029585799); draw(A--B--C--cycle); draw(B--D); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NE); [/asy]

$\textrm{(A)} \ 9 \qquad \textrm{(B)} \ 10 \qquad \textrm{(C)} \ 11 \qquad \textrm{(D)} \ 12 \qquad \textrm{(E)} \ 13$

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

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Category: AHSME
Points: 2
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