document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=178210>Question 178210</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In each of the following, determine the number of sides of<BR>\n" );
document.write( "a regular polygon with the stated property. If such a regular<BR>\n" );
document.write( "polygon does not exist, explain why.<BR>\n" );
document.write( "Each exterior angle measures 25°.<BR>\n" );
document.write( "The total number of diagonals is 4860. </SPAN>\n" );
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document.write( "Each exterior angle measures 25°\r<BR>\n" );
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document.write( "The sum of the exterior angles of any polygon is 360°.  Therefore if a regular polygon has exterior angles of x°, then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large {360 \over x}\"> must be an integer.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large {360 \over 25} = 14.4\">  Not an integer, therefore there is no regular polygon with exterior angles measuring 25°\r<BR>\n" );
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document.write( "The number of diagonals of a polygon is given by\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large {n(n - 3) \over 2}\">\r<BR>\n" );
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document.write( "You are given that there are 4860 diagonals, so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large {n(n - 3) \over 2} = 4860\">\r<BR>\n" );
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document.write( "Multiply by 2, remove parentheses, and put the equation in standard quadratic form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n^2 - 3n - 9720 = 0\">\r<BR>\n" );
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document.write( "Applying the quadratic formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n = \frac {-(-3) \pm sqrt{(-3)^2 - 4(1)(-9720)}}{2(1)} = \frac {3 \pm sqrt {38889}}{2} = \frac {3 \pm 3 sqrt{4321}}{2}\">\r<BR>\n" );
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document.write( "But one of these root is a negative number and both of them are irrational.  Therefore there is no regular polygon with 4860 diagonals.\r<BR>\n" );
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