document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1147941>Question 1147941</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The length of the diagonal of a rectangle is shorter than the semi-perimeter by 1/3 the length of the longer side. If the dimensions of all sides are integers, find the minimum length of the shorter side.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #769291 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=769291\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let x be the length of the short side, and let y be the length of the long side.<br><BR>\n" );
document.write( "The diagonal is then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2%2By%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2%2By%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "The problem says the diagonal is less than the semi-perimeter by 1/3 the length of the longer side; that is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2By%29-%281%2F3%29y+=+x%2B%282%2F3%29y ALIGN=MIDDLE  ALT=\"%28x%2By%29-%281%2F3%29y+=+x%2B%282%2F3%29y\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  So<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2%2By%5E2%29+=+x%2B%282%2F3%29y ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2%2By%5E2%29+=+x%2B%282%2F3%29y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asqrt%28x%5E2%2By%5E2%29+=+3x%2B2y ALIGN=MIDDLE  ALT=\"3%2Asqrt%28x%5E2%2By%5E2%29+=+3x%2B2y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2%2B9y%5E2+=+9x%5E2%2B12xy%2B4y%5E2 ALIGN=MIDDLE  ALT=\"9x%5E2%2B9y%5E2+=+9x%5E2%2B12xy%2B4y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y%5E2-12xy+=+0 ALIGN=MIDDLE  ALT=\"5y%5E2-12xy+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%285y-12x%29+=+0 ALIGN=MIDDLE  ALT=\"y%285y-12x%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "y=0 would make no sense in the problem, so 5y-12x = 0.  Then, if x and y are integers, x=5k and y=12k for some positive integer k.  And since we are looking for the minimum length of the shorter side that satisfies the conditions, x=5 and y=12.<br><BR>\n" );
document.write( "ANSWER: The minimum length of the shorter side is 5.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );