document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205030>Question 1205030</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given 30-60-90 triangle with sides p\sqrt(6) , p\sqrt(2  ), q\sqrt(3); find p, q<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #841626 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13215)</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=841626\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Don't use \"\\" -- it might have a special meaning in some area(s) of mathematics.<br><BR>\n" );
document.write( "I assume the side lengths are p/sqrt(6), q/sqrt(3), and p/sqrt(2).<br><BR>\n" );
document.write( "The squares of the side lengths are then p^2/6, q^2/3, and p^2/2.<br><BR>\n" );
document.write( "On first glance, with the denominators 6, 3, and 2, I immediately see that, if p and q are both equal to the same number x, then I have x^2/6+x^2/3=x^2/2, which is true for all values of x.<br><BR>\n" );
document.write( "So the problem has an infinite number of solutions in which p=q.<br><BR>\n" );
document.write( "But there might be other solutions hiding somewhere, so lets' look at the problem more closely.<br><BR>\n" );
document.write( "We know that p/sqrt(2) is greater than p/sqrt(6); but q/sqrt(3) could be less than or greater than p/sqrt(2).  So there are two cases to consider: the longest side (hypotenuse) can be either p/sqrt(2) or q/sqrt(3).<br><BR>\n" );
document.write( "Case 1: the hypotenuse is p/sqrt(2)<br><BR>\n" );
document.write( "(Note this is the case discussed informally above.)<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28p%2Fsqrt%286%29%29%5E2%2B%28q%2Fsqrt%283%29%29%5E2=%28p%2Fsqrt%282%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28p%2Fsqrt%286%29%29%5E2%2B%28q%2Fsqrt%283%29%29%5E2=%28p%2Fsqrt%282%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5E2%2F6%2Bq%5E2%2F3=p%5E2%2F2 ALIGN=MIDDLE  ALT=\"p%5E2%2F6%2Bq%5E2%2F3=p%5E2%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5E2%2B2q%5E2=3p%5E2 ALIGN=MIDDLE  ALT=\"p%5E2%2B2q%5E2=3p%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2q%5E2=2p%5E2 ALIGN=MIDDLE  ALT=\"2q%5E2=2p%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5E2=q%5E2 ALIGN=MIDDLE  ALT=\"p%5E2=q%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=q ALIGN=MIDDLE  ALT=\"p=q\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Case 2: The hypotenuse is q/sqrt(3)<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28p%2Fsqrt%282%29%29%5E2%2B%28p%2Fsqrt%286%29%29%5E2=%28q%2Fsqrt%283%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28p%2Fsqrt%282%29%29%5E2%2B%28p%2Fsqrt%286%29%29%5E2=%28q%2Fsqrt%283%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5E2%2F2%2Bp%5E2%2F6=q%5E2%2F3 ALIGN=MIDDLE  ALT=\"p%5E2%2F2%2Bp%5E2%2F6=q%5E2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3p%5E2%2Bp%5E2=2q%5E2 ALIGN=MIDDLE  ALT=\"3p%5E2%2Bp%5E2=2q%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4p%5E2=2q%5E2 ALIGN=MIDDLE  ALT=\"4p%5E2=2q%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2p%5E2=q%5E2 ALIGN=MIDDLE  ALT=\"2p%5E2=q%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Asqrt%282%29=q ALIGN=MIDDLE  ALT=\"p%2Asqrt%282%29=q\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This case also has an infinite number of solutions, where p is any number x and q is x*sqrt(2).<br><BR>\n" );
document.write( "ANSWER:<BR>\n" );
document.write( "p = any number;<BR>\n" );
document.write( "q = p OR q=p*sqrt(2)<br><BR>\n" );
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