document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1041968>Question 1041968</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The nine-pointed star shown here has a regular nonagon at its center. What is the measure of each angle at the tips of the star to the nearest .1°? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #656985 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=656985\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I do not see your star, but I think I can make three possible stars by extending the sides of the nonagon.  -0.026,2.82<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28645%2C300%2C-6.6%2C2%2C-2%2C2%2C%0D%0Aline%281%2C0%2C0.766%2C0.643%29%2Cline%280.766%2C0.643%2C0.174%2C0.985%29%2Cline%280.174%2C0.985%2C-0.5%2C0.866%29%2C%0D%0Aline%28-0.5%2C0.866%2C-0.94%2C0.342%29%2Cline%28-0.94%2C0.342%2C-0.94%2C-0.342%29%2Cline%28-0.94%2C-0.342%2C-0.5%2C-0.866%29%2C%0D%0Aline%28-0.5%2C-0.866%2C0.174%2C-0.985%29%2Cline%280.174%2C-0.985%2C0.766%2C-0.643%29%2Cline%280.766%2C-0.643%2C1%2C0%29%2C%0D%0Agreen%28arrow%281%2C0%2C2.026%2C-2.819%29%29%2Cgreen%28arrow%280.766%2C0.643%2C-0.26%2C3.462%29%29%2C%0D%0Agreen%28arrow%280.766%2C0.643%2C3.364%2C-0.857%29%29%2Cgreen%28arrow%280.174%2C0.985%2C-2.424%2C2.485%29%29%2C%0D%0Agreen%28arrow%280.174%2C0.985%2C3.128%2C1.506%29%29%2Cgreen%28arrow%28-0.5%2C0.866%2C-8.378%2C-0.523%29%29%2C%0D%0Agreen%28arrow%28-0.5%2C0.866%2C1.428%2C3.164%29%29%2Cgreen%28arrow%28-0.94%2C0.342%2C-3.511%2C-2.722%29%29%2C%0D%0Agreen%28arrow%28-0.94%2C0.342%2C-0.94%2C3.342%29%29%2Cgreen%28arrow%28-0.94%2C-0.342%2C-0.94%2C-3.342%29%29%2C%0D%0Agreen%28arrow%28-0.94%2C-0.342%2C-3.511%2C2.722%29%29%2Cgreen%28arrow%28-0.5%2C-0.866%2C1.428%2C-3.164%29%29%2C%0D%0Agreen%28arrow%28-0.5%2C-0.866%2C-8.378%2C0.523%29%29%2Cgreen%28arrow%280.174%2C-0.985%2C3.128%2C-1.506%29%29%2C%0D%0Agreen%28arrow%280.174%2C-0.985%2C-2.424%2C-2.485%29%29%2Cgreen%28arrow%280.766%2C-0.643%2C3.364%2C0.857%29%29%2C%0D%0Agreen%28arrow%280.766%2C-0.643%2C-0.26%2C-3.462%29%29%2Cgreen%28arrow%281%2C0%2C2.026%2C2.819%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For an easy problem you would have one of the 3 possible stars: the one that is made by adding isosceles triangles built around the nonagon. Each one of those isosceles triangles has a side of the nonagon as its base.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28645%2C300%2C-6.6%2C2%2C-2%2C2%2C%0D%0Agreen%28arrow%281%2C0%2C2.026%2C-2.819%29%29%2Cgreen%28arrow%280.766%2C0.643%2C-0.26%2C3.462%29%29%2C%0D%0Agreen%28arrow%280.766%2C0.643%2C3.364%2C-0.857%29%29%2Cgreen%28arrow%280.174%2C0.985%2C-2.424%2C2.485%29%29%2C%0D%0Agreen%28arrow%280.174%2C0.985%2C3.128%2C1.506%29%29%2Cgreen%28arrow%28-0.5%2C0.866%2C-8.378%2C-0.523%29%29%2C%0D%0Agreen%28arrow%28-0.5%2C0.866%2C1.428%2C3.164%29%29%2Cgreen%28arrow%28-0.94%2C0.342%2C-3.511%2C-2.722%29%29%2C%0D%0Agreen%28arrow%28-0.94%2C0.342%2C-0.94%2C3.342%29%29%2Cgreen%28arrow%28-0.94%2C-0.342%2C-0.94%2C-3.342%29%29%2C%0D%0Agreen%28arrow%28-0.94%2C-0.342%2C-3.511%2C2.722%29%29%2Cgreen%28arrow%28-0.5%2C-0.866%2C1.428%2C-3.164%29%29%2C%0D%0Agreen%28arrow%28-0.5%2C-0.866%2C-8.378%2C0.523%29%29%2Cgreen%28arrow%280.174%2C-0.985%2C3.128%2C-1.506%29%29%2C%0D%0Agreen%28arrow%280.174%2C-0.985%2C-2.424%2C-2.485%29%29%2Cgreen%28arrow%280.766%2C-0.643%2C3.364%2C0.857%29%29%2C%0D%0Agreen%28arrow%280.766%2C-0.643%2C-0.26%2C-3.462%29%29%2Cgreen%28arrow%281%2C0%2C2.026%2C2.819%29%29%2C%0D%0Atriangle%281%2C0%2C1.153%2C-0.42%2C0.766%2C-0.643%29%2C%0D%0Atriangle%280.766%2C0.643%2C1.153%2C0.42%2C1%2C0%29%2C%0D%0Atriangle%280.174%2C0.985%2C0.613%2C1.062%2C0.766%2C0.643%29%2C%0D%0Atriangle%28-0.5%2C0.866%2C-0.213%2C1.208%2C0.174%2C0.985%29%2C%0D%0Atriangle%28-0.94%2C0.342%2C-0.94%2C0.789%2C-0.5%2C0.866%29%2C%0D%0Atriangle%28-0.94%2C-0.342%2C-1.227%2C0%2C-0.94%2C0.342%29%2C%0D%0Atriangle%28-0.5%2C-0.866%2C-0.94%2C-0.788%2C-0.94%2C-0.342%29%2C%0D%0Atriangle%280.174%2C-0.985%2C-0.213%2C-1.208%2C-0.5%2C-0.866%29%2C%0D%0Atriangle%280.766%2C-0.643%2C0.613%2C-1.062%2C0.174%2C-0.985%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The two base angles of each of those isosceles triangles are exterior angles of the nonagon.<BR>\n" );
document.write( "The measure of an exterior angle of a regular 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 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo%2Fn ALIGN=MIDDLE  ALT=\"360%5Eo%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the measure of an exterior angle of a regular nonagon is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo%2F9=40%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo%2F9=40%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, those isosceles triangles have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > base angles measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40%5Eo ALIGN=MIDDLE  ALT=\"40%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The other angle is at a tip of the start.<BR>\n" );
document.write( "Since the sum of the measures of the angles of any triangle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the angle at the tip og the star measures<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo-2%2A40%5Eo=180%5Eo-80%5Eo=highlight%28100%5Eo%29 ALIGN=MIDDLE  ALT=\"180%5Eo-2%2A40%5Eo=180%5Eo-80%5Eo=highlight%28100%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "The other two stars have sharper tips, whose angles are not quite as easy to calculate.<BR>\n" );
document.write( "Here is the smaller one:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28645%2C300%2C-6.6%2C2%2C-2%2C2%2C%0D%0Aline%281%2C0%2C0.766%2C0.643%29%2Cline%280.766%2C0.643%2C0.174%2C0.985%29%2Cline%280.174%2C0.985%2C-0.5%2C0.866%29%2C%0D%0Aline%28-0.5%2C0.866%2C-0.94%2C0.342%29%2Cline%28-0.94%2C0.342%2C-0.94%2C-0.342%29%2Cline%28-0.94%2C-0.342%2C-0.5%2C-0.866%29%2C%0D%0Aline%28-0.5%2C-0.866%2C0.174%2C-0.985%29%2Cline%280.174%2C-0.985%2C0.766%2C-0.643%29%2Cline%280.766%2C-0.643%2C1%2C0%29%2C%0D%0Agreen%28arrow%281%2C0%2C2.026%2C-2.819%29%29%2Cgreen%28arrow%280.766%2C0.643%2C-0.26%2C3.462%29%29%2C%0D%0Agreen%28arrow%280.766%2C0.643%2C3.364%2C-0.857%29%29%2Cgreen%28arrow%280.174%2C0.985%2C-2.424%2C2.485%29%29%2C%0D%0Agreen%28arrow%280.174%2C0.985%2C3.128%2C1.506%29%29%2Cgreen%28arrow%28-0.5%2C0.866%2C-8.378%2C-0.523%29%29%2C%0D%0Agreen%28arrow%28-0.5%2C0.866%2C1.428%2C3.164%29%29%2Cgreen%28arrow%28-0.94%2C0.342%2C-3.511%2C-2.722%29%29%2C%0D%0Agreen%28arrow%28-0.94%2C0.342%2C-0.94%2C3.342%29%29%2Cgreen%28arrow%28-0.94%2C-0.342%2C-0.94%2C-3.342%29%29%2C%0D%0Agreen%28arrow%28-0.94%2C-0.342%2C-3.511%2C2.722%29%29%2Cgreen%28arrow%28-0.5%2C-0.866%2C1.428%2C-3.164%29%29%2C%0D%0Agreen%28arrow%28-0.5%2C-0.866%2C-8.378%2C0.523%29%29%2Cgreen%28arrow%280.174%2C-0.985%2C3.128%2C-1.506%29%29%2C%0D%0Agreen%28arrow%280.174%2C-0.985%2C-2.424%2C-2.485%29%29%2Cgreen%28arrow%280.766%2C-0.643%2C3.364%2C0.857%29%29%2C%0D%0Agreen%28arrow%280.766%2C-0.643%2C-0.26%2C-3.462%29%29%2Cgreen%28arrow%281%2C0%2C2.026%2C2.819%29%29%2C%0D%0Ared%28line%280.612%2C1.066%2C0.326%2C1.851%29%29%2Cred%28line%28-0.211%2C1.211%2C0.326%2C1.851%29%29%2C%0D%0Ared%28line%28-0.216%2C1.21%2C-0.94%2C1.628%29%29%2Cred%28line%28-0.94%2C0.792%2C-0.94%2C1.628%29%29%2C%0D%0Ared%28line%28-1.229%2C0.003%2C-1.766%2C0.643%29%29%2Cred%28line%28-0.943%2C0.788%2C-1.766%2C0.643%29%29%2C%0D%0Ared%28line%28-0.943%2C-0.788%2C-1.766%2C-0.643%29%29%2Cred%28line%28-1.229%2C-0.003%2C-1.766%2C-0.643%29%29%2C%0D%0Ared%28line%28-0.94%2C-0.792%2C-0.94%2C-1.628%29%29%2Cred%28line%28-0.216%2C-1.21%2C-0.94%2C-1.628%29%29%2C%0D%0Ared%28line%280.612%2C-1.066%2C0.326%2C-1.851%29%29%2Cred%28line%28-0.211%2C-1.211%2C0.326%2C-1.851%29%29%2C%0D%0Ared%28line%281.154%2C-0.423%2C1.44%2C-1.208%29%29%2Cred%28line%280.617%2C-1.063%2C1.44%2C-1.208%29%29%2C%0D%0Ared%28line%281.156%2C0.418%2C1.88%2C0%29%29%2Cred%28line%281.156%2C-0.418%2C1.88%2C0%29%29%2C%0D%0Ared%28line%280.617%2C1.063%2C1.44%2C1.208%29%29%2Cred%28line%281.154%2C0.423%2C1.44%2C1.208%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The larger one, with the sharpest tips is too big A partial drawing is shown below.<BR>\n" );
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document.write( "</TD></TR></TABLE>\n" );