<p>Consider all <a href="https://artofproblemsolving.com/wiki/index.php/Triangle" title="Triangle">triangles</a> <img alt="$ABC$" height="13" src="https://latex.artofproblemsolving.com/e/2/a/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png" width="42" /> satisfying in the following conditions: <img alt="$AB = AC$" height="13" src="https://latex.artofproblemsolving.com/f/3/a/f3ac37c9fa391eebb9601bfaca8bfbfaabdf5ca1.png" width="80" />, <img alt="$D$" height="14" src="https://latex.artofproblemsolving.com/9/f/f/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png" width="17" /> is a point on <img alt="$\overline{AC}$" height="18" src="https://latex.artofproblemsolving.com/1/2/5/12517bbc31cb9026c71299b7a0d232d07e60f284.png" width="30" /> for which <img alt="$\overline{BD} \perp \overline{AC}$" height="15" src="https://latex.artofproblemsolving.com/6/3/5/635aaf493927a2a6c0c85d1613ff38840fb6ba42.png" width="82" />, <img alt="$AC$" height="14" src="https://latex.artofproblemsolving.com/a/1/7/a179ff2638e4799cadd820db205c2beff6299ce9.png" width="30" /> and <img alt="$CD$" height="12" src="https://latex.artofproblemsolving.com/0/9/4/0948661e822f2a953c43a57ac9e40b2734476de4.png" width="29" /> are integers, and <img alt="$BD^{2} = 57$" height="15" src="https://latex.artofproblemsolving.com/2/3/7/237ec66e204854caf0ae63318e6deec53e0a4a2c.png" width="80" />. Among all such triangles, the smallest possible value of <img alt="$AC$" height="14" src="https://latex.artofproblemsolving.com/a/1/7/a179ff2638e4799cadd820db205c2beff6299ce9.png" width="30" /> is</p> <p><img alt="[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]" height="252" src="https://latex.artofproblemsolving.com/4/d/0/4d08273e2a3ddb2c8fffd607582fa89298ab91db.png" width="218" /></p>

Question No: 252

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$",D,NE); [/asy]$

Editorial

Tags: Algebra Geometry

Question Details