document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=478757>Question 478757</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider a polygon with n sides.a)Find a mathematical equation that relates the number of diagonals in the polygon to the number of sides.b)Use your equation to find, 1)The number of diagonals if the polygon has 17 sides.2)The number of sides if the polygon has 119 diagonals. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #328090 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=328090\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the formula for determining the number of diagonals in a polygon 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 derivation of this formula is as follows:<BR>\n" );
document.write( "each vertex in a polygon can connect to n-3 other vertices.<BR>\n" );
document.write( "the only ones it can't connect to are itself and the vertices that are adjacent to it.<BR>\n" );
document.write( "if you have a pentagon, there are 5 vertices.  each one of these can connect to 2 other vertices for a total of 10 diagonals.  each diagonal connects between 2 vertices, so the number of diagonals has to be divided by 2 in order to avoid double counting.<BR>\n" );
document.write( "the number of diagonals in the pentagon is therefore (5 * 2) / 2 = 5<BR>\n" );
document.write( "a picture of your pentagon is shown below.<BR>\n" );
document.write( "explanations are below the picture.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/problems/pentagon_1.jpg\" alt = \"$$$$\"/ ><BR>\n" );
document.write( "point A connects to C and D<BR>\n" );
document.write( "point B connects to D and E<BR>\n" );
document.write( "point C connects to E and A<BR>\n" );
document.write( "point D connects to A and B<BR>\n" );
document.write( "point E connects to B and C<BR>\n" );
document.write( "the diagonals formed are:<BR>\n" );
document.write( "AC<BR>\n" );
document.write( "AD<BR>\n" );
document.write( "BD<BR>\n" );
document.write( "BE<BR>\n" );
document.write( "CE<BR>\n" );
document.write( "CA<BR>\n" );
document.write( "DA<BR>\n" );
document.write( "DB<BR>\n" );
document.write( "EB<BR>\n" );
document.write( "EC<BR>\n" );
document.write( "that's a total of 10.<BR>\n" );
document.write( "however,<BR>\n" );
document.write( "AC and CA are the same diagonal<BR>\n" );
document.write( "AD and DA are the same diagonal<BR>\n" );
document.write( "BD and DB are the same diagonal<BR>\n" );
document.write( "BE and EB are the same diagonal<BR>\n" );
document.write( "CE and EC are the same diagonal<BR>\n" );
document.write( "cross off the duplicates and you're left with 5.<BR>\n" );
document.write( "that's why the formula for determining the number of diagonals is:<BR>\n" );
document.write( "(n*(n-3))/2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );