document.write( "Question 1209330: Given regular heptagon ABCDEFG, a circle can be drawn that is tangent to DC at C and to EF at F. What is radius of the circle if the side length of the heptagon is 1? \n" ); document.write( "

Algebra.Com's Answer #848381 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        The solution in the post by @ElectricPavlov is incorrect.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        It is incorrect, since it uses \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "                \"the distance CF = side length of the heptagon = 1\"   in n.8,\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        which is FATALLY WRONG.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "The side of this regular heptagon is 1 (given).  \r\n" );
document.write( "\r\n" );
document.write( "Let O be the center of the heptagon ABCDEFG.\r\n" );
document.write( "\r\n" );
document.write( "Let the radius of the circumscribed circle around the heptagon be r\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Its central angle is a =  = 51.4286 degrees.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "For the radius r we have this equation\r\n" );
document.write( "\r\n" );
document.write( "    r*sin(a/2) = 1/2,  which gives  r =  =  = 1.1524.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now consider triangle OCF.  It is isosceles triangle.\r\n" );
document.write( "\r\n" );
document.write( "Its lateral sides OC and OF have the length r, and they conclude the angle COF of 3a =  = 154.2857 degrees.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the length of CF is (use the cosine law for triangle OCF)\r\n" );
document.write( "\r\n" );
document.write( "    |CF| =  =  =  = 2.247.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Let O' be the center of the circle, which touches  CD at C  and  touches EF at F.\r\n" );
document.write( "\r\n" );
document.write( "Let R be the radius of this circle, which the problem asks to determine.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The angle at O' between perpendiculars to CD at C  and  to EF at F is 2a.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now apply the cosine law to triangle  O'CF\r\n" );
document.write( "\r\n" );
document.write( "    R^2 + R^2 - 2R*R*cos(2a) = |CF|^2\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "and find\r\n" );
document.write( "\r\n" );
document.write( "    R =  =  =  = 1.4354.    ANSWER\r\n" );
document.write( "\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "It is how the problem SHOULD be solved, if to do it correctly.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );