document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=652123>Question 652123</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what polygon has a number of sides that is five times the number of diagonals </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #407920 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=407920\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The way I understand geometry, the answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28NONE%29 ALIGN=MIDDLE  ALT=\"highlight%28NONE%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "I am betting there is a mix-up in the wording of the problem.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The way I understand geometry,<BR>\n" );
document.write( "a diagonal of a polygon is a segment that connects two vertices of the polygon, but is not a side of the polygon.<BR>\n" );
document.write( "If a polygon has <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, it will have n vertices.<BR>\n" );
document.write( "Each vertex will be connected to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > other vertices by sides,<BR>\n" );
document.write( "and can be connected to the remaining <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-3 ALIGN=MIDDLE  ALT=\"n-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > other vertices by a diagonal.<BR>\n" );
document.write( "If we multiply the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-3 ALIGN=MIDDLE  ALT=\"n-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > diagonals coming out of each vertex times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the number of vertices,<BR>\n" );
document.write( "we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-3%29 ALIGN=MIDDLE  ALT=\"n%28n-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but we are counting each diagonal twice.<BR>\n" );
document.write( "The 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 has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=n%28n-3%29%2F2 ALIGN=MIDDLE  ALT=\"d=n%28n-3%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > diagonals.<BR>\n" );
document.write( " <BR>\n" );
document.write( "To find what polygon has a number of sides that is five times the number of diagonals, we can set up an equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=5%2A%28n%28n-3%29%2F2%29 ALIGN=MIDDLE  ALT=\"n=5%2A%28n%28n-3%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or we can make a table comparing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
document.write( "\n" );
document.write( "For that table:<BR>\n" );
document.write( "A triangle has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > sides and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > diagonals.<BR>\n" );
document.write( "A quadrilateral has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > sides and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > diagonals.<BR>\n" );
document.write( "From then on, the number of sides is equal to or less than the number of diagonals.<BR>\n" );
document.write( "A pentagon has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > sides and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > diagonals.<BR>\n" );
document.write( "A hexagon has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > sides and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > diagonals.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Either way (equation or table), the answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28NONE%29 ALIGN=MIDDLE  ALT=\"highlight%28NONE%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . <BR>\n" );
document.write( " <BR>\n" );
document.write( "If we wanted to know what polygon has a number of diagonals that is five times the number of sides, our equation would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5n=n%28n-3%29%2F2 ALIGN=MIDDLE  ALT=\"5n=n%28n-3%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5n=n%28n-3%29%2F2 ALIGN=MIDDLE  ALT=\"5n=n%28n-3%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A5n=2%2A%28n%28n-3%29%2F2%29 ALIGN=MIDDLE  ALT=\"2%2A5n=2%2A%28n%28n-3%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10n=n%28n-3%29 ALIGN=MIDDLE  ALT=\"10n=n%28n-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10n=n%5E2-3n%29 ALIGN=MIDDLE  ALT=\"10n=n%5E2-3n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10n%2B3n=n%5E2-3n%2B3n%29 ALIGN=MIDDLE  ALT=\"10n%2B3n=n%5E2-3n%2B3n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13n=n%5E2%29 ALIGN=MIDDLE  ALT=\"13n=n%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=0 ALIGN=MIDDLE  ALT=\"n=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not make a polygon, we know <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13n=n%5E2%29 ALIGN=MIDDLE  ALT=\"13n=n%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13n%2Fn=n%5E2%2Fn%29 ALIGN=MIDDLE  ALT=\"13n%2Fn=n%5E2%2Fn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13=n%29 ALIGN=MIDDLE  ALT=\"13=n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28n=13%29 ALIGN=MIDDLE  ALT=\"highlight%28n=13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );