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 #848379 by ElectricPavlov(122)\"\" \"About 
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Certainly, let's find the radius of the circle.\r
\n" ); document.write( "\n" ); document.write( "**1. Find the interior angle of the heptagon:**\r
\n" ); document.write( "\n" ); document.write( "* The interior angle of a regular heptagon is given by:
\n" ); document.write( " * (n - 2) * 180° / n
\n" ); document.write( " * where n is the number of sides (n = 7)
\n" ); document.write( " * Interior angle = (7 - 2) * 180° / 7 = 900° / 7 ≈ 128.57°\r
\n" ); document.write( "\n" ); document.write( "**2. Find the angle at the center of the heptagon:**\r
\n" ); document.write( "\n" ); document.write( "* The central angle of a regular heptagon is given by:
\n" ); document.write( " * 360° / n
\n" ); document.write( " * Central angle = 360° / 7 ≈ 51.43°\r
\n" ); document.write( "\n" ); document.write( "**3. Find the angle ∠DCF:**\r
\n" ); document.write( "\n" ); document.write( "* ∠DCF = 2 * (Interior angle) - 360°
\n" ); document.write( " * ∠DCF = 2 * 128.57° - 360° = 257.14° - 360° = -102.86°
\n" ); document.write( " * Since we're dealing with angles on a circle, we can consider ∠DCF = 360° - 102.86° = 257.14°\r
\n" ); document.write( "\n" ); document.write( "**4. Find the angle ∠CDF:**\r
\n" ); document.write( "\n" ); document.write( "* ∠CDF = 180° - Interior angle = 180° - 128.57° = 51.43°\r
\n" ); document.write( "\n" ); document.write( "**5. Construct the circle:**\r
\n" ); document.write( "\n" ); document.write( "* Draw the circle tangent to DC at C and EF at F.
\n" ); document.write( "* Let O be the center of the circle.
\n" ); document.write( "* Let R be the radius of the circle.\r
\n" ); document.write( "\n" ); document.write( "**6. Find the distance OC:**\r
\n" ); document.write( "\n" ); document.write( "* In triangle ODC, ∠OCD = 90° (tangent to the circle)
\n" ); document.write( "* OC = R (radius of the circle)\r
\n" ); document.write( "\n" ); document.write( "**7. Find the distance OF:**\r
\n" ); document.write( "\n" ); document.write( "* In triangle OEF, ∠OFE = 90° (tangent to the circle)
\n" ); document.write( "* OF = R (radius of the circle)\r
\n" ); document.write( "\n" ); document.write( "**8. Find the distance CF:**\r
\n" ); document.write( "\n" ); document.write( "* CF = side length of the heptagon = 1\r
\n" ); document.write( "\n" ); document.write( "**9. Use the Law of Cosines in triangle OCF:**\r
\n" ); document.write( "\n" ); document.write( "* CF² = OC² + OF² - 2 * OC * OF * cos(∠DCF)
\n" ); document.write( "* 1² = R² + R² - 2 * R * R * cos(257.14°)
\n" ); document.write( "* 1 = 2R² - 2R² * cos(257.14°)
\n" ); document.write( "* 1 = 2R² * (1 - cos(257.14°))
\n" ); document.write( "* R² = 1 / [2 * (1 - cos(257.14°))]
\n" ); document.write( "* R = √[1 / [2 * (1 - cos(257.14°))]]
\n" ); document.write( "* R ≈ 0.4339\r
\n" ); document.write( "\n" ); document.write( "**Therefore, the radius of the circle is approximately 0.4339.**\r
\n" ); document.write( "\n" ); document.write( "**Note:**\r
\n" ); document.write( "\n" ); document.write( "* This calculation assumes that the circle is externally tangent to both DC and EF.
\n" ); document.write( "* If the circle is internally tangent to one of the sides, the calculation would be different.
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