document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=823339>Question 823339</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the  set of polygons in which the number of diagonals is greater than the sum of the measures of the angles. Please help me figure this out. I've tried everything. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #495493 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=495493\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I assume it is the sum of the measures of the interior angles, in degrees.<BR>\n" );
document.write( "If you measure the angles in radians, there is no solution.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The number of diagonals in a polygon with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > sides is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-3%29%2F2 ALIGN=MIDDLE  ALT=\"n%28n-3%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The sum of the interior angles, in degrees, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%28n-2%29 ALIGN=MIDDLE  ALT=\"180%28n-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The inequality to solve is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-3%29%2F2%3E180%28n-2%29 ALIGN=MIDDLE  ALT=\"n%28n-3%29%2F2%3E180%28n-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Solving:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-3%29%2F2%3E180%28n-2%29 ALIGN=MIDDLE  ALT=\"n%28n-3%29%2F2%3E180%28n-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-3%29%3E360%28n-2%29 ALIGN=MIDDLE  ALT=\"n%28n-3%29%3E360%28n-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2-3n%3E360n-720 ALIGN=MIDDLE  ALT=\"n%5E2-3n%3E360n-720\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2-3n-360n%2B720%3E0 ALIGN=MIDDLE  ALT=\"n%5E2-3n-360n%2B720%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2-363n%2B720%3E0 ALIGN=MIDDLE  ALT=\"n%5E2-363n%2B720%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28n%29=n%5E2-363n%2B720%3E0 ALIGN=MIDDLE  ALT=\"P%28n%29=n%5E2-363n%2B720%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a quadratic function.<BR>\n" );
document.write( "It graphs as a parabola with a minimum.<BR>\n" );
document.write( "It may be negative for a certain interval of intermediate values,<BR>\n" );
document.write( "but outside of that middle interval it will be positive, so we will have some solutions.<BR>\n" );
document.write( "We know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%29=720%3E0 ALIGN=MIDDLE  ALT=\"P%280%29=720%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but we want to know the solutions for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3E=3 ALIGN=MIDDLE  ALT=\"n%3E=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "because a polygon has to have at least 3 sides.<BR>\n" );
document.write( "If we find the solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28n%29=n%5E2-363n%2B720=0 ALIGN=MIDDLE  ALT=\"P%28n%29=n%5E2-363n%2B720=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , if any,<BR>\n" );
document.write( "those solutions will be the ends of that middle interval where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28n%29%3C0 ALIGN=MIDDLE  ALT=\"P%28n%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <BR>\n" );
document.write( "we will have our answer.<BR>\n" );
document.write( "Applying the quadratic formula to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2-363n%2B720=0 ALIGN=MIDDLE  ALT=\"n%5E2-363n%2B720=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > we find<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28363+%2B-+sqrt%28363%5E2-4%2A1%2A720+%29%29%2F%282%2A1%29=%28363+%2B-+sqrt%28131769-2880%29%29%2F2=%28363+%2B-+sqrt%28128889%29%29%2F2+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The approximate values are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5B1%5D=1.995 ALIGN=MIDDLE  ALT=\"n%5B1%5D=1.995\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5B2%5D=361.005 ALIGN=MIDDLE  ALT=\"n%5B2%5D=361.005\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For the number of diagonals to be greater than the sum of the measures of the interior angles in degrees, we need<BR>\n" );
document.write( "either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3Cn%5B1%5D ALIGN=MIDDLE  ALT=\"n%3Cn%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3En%5B2%5D ALIGN=MIDDLE  ALT=\"n%3En%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3Cn%5B1%5D ALIGN=MIDDLE  ALT=\"n%3Cn%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives no polygon solution.<BR>\n" );
document.write( "The inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3En%5B2%5D ALIGN=MIDDLE  ALT=\"n%3En%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > means that all polygons with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28n%3E=362%29 ALIGN=MIDDLE  ALT=\"highlight%28n%3E=362%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (362 or more sides) will have<BR>\n" );
document.write( "a number of diagonals that is greater than the sum of the measures of the interior angles in degrees. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );