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 #848704 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Since no one has drawn the figure, and also has not given an exact value\r\n" );
document.write( "for the radius, I thought I would do so, with an exact solution in terms \r\n" );
document.write( "of trigonometric values.\r\n" );
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document.write( "We draw perpendiculars to DC at C and to EF at F, and they must intersect\r\n" );
document.write( "at the center of the circle.  We also draw FC to make an isosceles trapezoid\r\n" );
document.write( "from which we can get the measurements for the angles, in particular angle CFE,\r\n" );
document.write( "which turns out to be \"2pi%2F7\" radians.  \r\n" );
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document.write( "The sum of the interior angles of a polygon with n-sides = \"%28n-2%29%2Api\"\r\n" );
document.write( "So each interior angle of the regular heptagon is \"expr%281%2F7%29%287-2%29%2Api=5pi%2F7\", so angles E and D are \"5pi%2F7\" each.\r\n" );
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document.write( "Isosceles trapezoid FEDC has sum of interior angles \"2pi\" so angles EFC and\r\n" );
document.write( "FCD are each \"expr%281%2F2%29%282pi-2%28%285pi%29%2F7%29%29\"\"%22%22=%22%22\"\"2pi%2F7\"\r\n" );
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document.write( "Next we'll draw in 3 perpendiculars to FC, namely EH, DJ, and OI.\r\n" );
document.write( "Notice that angles EFH and FOI have equal measures because they are both\r\n" );
document.write( "complements of the same angle IFO. Therefore angle FOI also measures \"2pi%2F7\".\r\n" );
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document.write( "Since each side of the heptagon is 1, \r\n" );
document.write( "(a) the hypotenuse EF of right triangle EFH is 1 and \"FH=cos%282pi%2F7%29\"\r\n" );
document.write( "(b) \"HI=1%2F2\" because HJ=ED=1 and HI is 1/2 of HJ\r\n" );
document.write( "(c) \"FI=FH%2BHI=cos%282pi%2F7%29%2B1%2F2\"\r\n" );
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document.write( "So from right triangle FIO, we can now find the desired radius FO\r\n" );
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document.write( "\"%28FI%29%2F%28FO%29=sin%282pi%2F7%29\" and \"FO=FI%5E%22%22%2Fsin%282pi%2F7%29=%28cos%282pi%2F7%29%2B1%2F2%29%2Fsin%282pi%2F7%29\"\r\n" );
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document.write( "So the exact answer for the radius is\r\n" );
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document.write( "\"radius+=+%28cos%282pi%2F7%29%2B1%2F2%29%2Fsin%282pi%2F7%29\" which can also be written as\r\n" );
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document.write( "\"cot%282pi%2F7%29%2Bexpr%281%2F2%29csc%282pi%2F7%29\"  <--EXACT SOLUTION!\r\n" );
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document.write( "That's approximately 1.436997393, which approximately agrees with Ikleyn's,\r\n" );
document.write( "and is even closer to Greenestamps', approximate solution. \r\n" );
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document.write( "Edwin
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