document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168530>Question 1168530</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A regular octagon is emclosed with a circle.it has an extension piece CDFE obtained by extendind F and C to E with BCE is a straight line.AF was the diameter of the circle. FE is a tangent to the circle at F.calclate the size of z(angle FEC) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851562 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851562\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this geometry problem step by step.\r<BR>\n" );
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document.write( "**1. Diagram and Definitions**\r<BR>\n" );
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document.write( "* We have a regular octagon inscribed in a circle.<BR>\n" );
document.write( "* AF is a diameter of the circle.<BR>\n" );
document.write( "* FC is extended to E, such that BCE is a straight line.<BR>\n" );
document.write( "* FE is a tangent to the circle at F.<BR>\n" );
document.write( "* We need to find the measure of angle FEC.\r<BR>\n" );
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document.write( "**2. Properties of a Regular Octagon**\r<BR>\n" );
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document.write( "* A regular octagon has 8 equal sides and 8 equal interior angles.<BR>\n" );
document.write( "* The measure of each interior angle of a regular octagon is:<BR>\n" );
document.write( "    * (n - 2) * 180° / n = (8 - 2) * 180° / 8 = 6 * 180° / 8 = 135°\r<BR>\n" );
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document.write( "**3. Angles in the Diagram**\r<BR>\n" );
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document.write( "* Let O be the center of the circle.<BR>\n" );
document.write( "* Since the octagon is regular, the central angle subtended by each side is 360° / 8 = 45°.<BR>\n" );
document.write( "* Angle AOF is 45 degrees.<BR>\n" );
document.write( "* Angle COF is 45 degrees.<BR>\n" );
document.write( "* Angle BOC is 45 degrees.<BR>\n" );
document.write( "* Since AF is a diameter, angle ACF is 135/2 = 67.5 degrees, because the angle ACF is half the interior angle of the octagon.<BR>\n" );
document.write( "* Angle AFC = 135/2 = 67.5 degrees.\r<BR>\n" );
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document.write( "**4. Tangent and Diameter Properties**\r<BR>\n" );
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document.write( "* Since FE is a tangent to the circle at F, angle AFE = 90°.\r<BR>\n" );
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document.write( "**5. Finding Angle CFE**\r<BR>\n" );
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document.write( "* We need to find angle CFE.<BR>\n" );
document.write( "* Since angle AFC = 67.5 degrees and angle AFE = 90 degrees.<BR>\n" );
document.write( "* angle CFE = angle AFE - angle AFC = 90 - 67.5 = 22.5 degrees.\r<BR>\n" );
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document.write( "**6. Finding Angle FEC**\r<BR>\n" );
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document.write( "* In triangle FEC, we need to find angle FEC.<BR>\n" );
document.write( "* Since BCE is a straight line, angle FCB = 180 - 135 = 45 degrees.<BR>\n" );
document.write( "* We know angle CFE = 22.5 degrees.<BR>\n" );
document.write( "* In triangle FEC, the sum of the angles is 180 degrees.<BR>\n" );
document.write( "* angle FEC + angle CFE + angle FCE = 180 degrees.<BR>\n" );
document.write( "* angle FEC + 22.5 + 45 = 180<BR>\n" );
document.write( "* angle FEC + 67.5 = 180<BR>\n" );
document.write( "* angle FEC = 180 - 67.5<BR>\n" );
document.write( "* angle FEC = 112.5 degrees.\r<BR>\n" );
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document.write( "**Therefore, the size of angle FEC is 112.5 degrees.**<BR>\n" );
document.write( " </SPAN>\n" );
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