document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1011941>Question 1011941</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the side of a regular octagon inscribed in a circle of radius 19 cm.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #627773 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=627773\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Think TRIANGLES. Triangles are the key to solving many problems.<BR>\n" );
document.write( "They are at the heart of geometry, trigonometry and engineering.<BR>\n" );
document.write( "Or think TRIGONOMETRY, if you wish.<BR>\n" );
document.write( "Use LAW OF COSINES, if that is what you were taught recently.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Here is the circle, with its center, the inscribed regular octagon,<BR>\n" );
document.write( "and some line segments that are at the same time<BR>\n" );
document.write( "diameters of the circle and diagonals of the octagon.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.1%2C1.1%2C-1.1%2C1.1%2C%0D%0Ared%28circle%280%2C0%2C1.01%29%29%2C%0D%0Acircle%280%2C0%2C0.02%29%2C%0D%0Atriangle%280%2C0%2C1%2C0%2C0.707%2C0.707%29%2C%0D%0Atriangle%280%2C0%2C0%2C1%2C0.707%2C0.707%29%2C%0D%0Atriangle%280%2C0%2C0%2C1%2C-0.707%2C0.707%29%2C%0D%0Atriangle%280%2C0%2C-1%2C0%2C-0.707%2C0.707%29%2C%0D%0Atriangle%280%2C0%2C1%2C0%2C0.707%2C-0.707%29%2C%0D%0Atriangle%280%2C0%2C0%2C-1%2C0.707%2C-0.707%29%2C%0D%0Atriangle%280%2C0%2C0%2C-1%2C-0.707%2C-0.707%29%2C%0D%0Atriangle%280%2C0%2C-1%2C0%2C-0.707%2C-0.707%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "The diagonals connecting opposite octagon vertices go through the center of the circle,<BR>\n" );
document.write( "and split the octagon into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > isosceles triangles.<BR>\n" );
document.write( "Each of those triangles has two sides measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19cm ALIGN=MIDDLE  ALT=\"19cm\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(the radius of the circle), and an angle measuring<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo%2F8=45%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo%2F8=45%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > (or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2F8=pi%2F4 ALIGN=MIDDLE  ALT=\"2pi%2F8=pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > if you prefer radians)<BR>\n" );
document.write( "at the center of the circle.<BR>\n" );
document.write( " <BR>\n" );
document.write( "USING TRIANGLES:<BR>\n" );
document.write( "Let's look at one of those triangles, and use the Pythagorean theorem<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-0.1%2C1.1%2C-0.2%2C1%2C%0D%0Atriangle%280%2C0%2C1%2C0%2C0.707%2C0.707%29%2C%0D%0Atriangle%280%2C0%2C0.707%2C0%2C0.707%2C0.707%29%2C%0D%0Arectangle%280.707%2C0%2C0.657%2C0.05%29%2C%0D%0Alocate%280.35%2C0.35%2C19%29%2C%0D%0Alocate%280.47%2C-0.02%2C19%29%2C%0D%0Aarrow%280.46%2C-0.05%2C0%2C-0.05%29%2C%0D%0Aarrow%280.54%2C-0.05%2C1%2C-0.05%29%2C%0D%0Alocate%280.34%2C0.06%2Cx%29%2C%0D%0Alocate%280.71%2C0.37%2Cx%29%2C%0D%0Alocate%280.83%2C0.06%2C19-x%29%2C%0D%0Alocate%280.86%2C0.37%2Cside%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx%5E2=19%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx%5E2=19%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2=19%5E2 ALIGN=MIDDLE  ALT=\"2x%5E2=19%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=19%5E2%2F2 ALIGN=MIDDLE  ALT=\"x%5E2=19%5E2%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=19%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"x=19%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the larger right triangle.<BR>\n" );
document.write( "For the smaller right triangle, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%2819-x%29%5E2=side%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B%2819-x%29%5E2=side%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We can substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"19%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19%5E2%2F2 ALIGN=MIDDLE  ALT=\"19%5E2%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > if you wish)<BR>\n" );
document.write( "sooner or later, as you wish.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%2819-x%29%5E2=side%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B%2819-x%29%5E2=side%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B19%5E2-2%2A19x%2Bx%5E2=side%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B19%5E2-2%2A19x%2Bx%5E2=side%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B19%5E2-2%2A19x=side%5E2 ALIGN=MIDDLE  ALT=\"2x%5E2%2B19%5E2-2%2A19x=side%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x%5E2-19x%29%2B19%5E2=side%5E2 ALIGN=MIDDLE  ALT=\"2%28x%5E2-19x%29%2B19%5E2=side%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2819%5E2%2F2-19%2A19%2Fsqrt%282%29%29%2B19%5E2=side%5E2 ALIGN=MIDDLE  ALT=\"2%2819%5E2%2F2-19%2A19%2Fsqrt%282%29%29%2B19%5E2=side%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19%5E2-19%5E2sqrt%282%29%2B19%5E2=side%5E2 ALIGN=MIDDLE  ALT=\"19%5E2-19%5E2sqrt%282%29%2B19%5E2=side%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A19%5E2-19%5E2sqrt%282%29=side%5E2 ALIGN=MIDDLE  ALT=\"2%2A19%5E2-19%5E2sqrt%282%29=side%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19%5E2%282-sqrt%282%29%29=side%5E2 ALIGN=MIDDLE  ALT=\"19%5E2%282-sqrt%282%29%29=side%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=side=sqrt%2819%5E2%282-sqrt%282%29%29%29 ALIGN=MIDDLE  ALT=\"side=sqrt%2819%5E2%282-sqrt%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28side=19sqrt%282-sqrt%282%29%29%29 ALIGN=MIDDLE  ALT=\"highlight%28side=19sqrt%282-sqrt%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or for an approximate measure <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28side=about14.54cm%29 ALIGN=MIDDLE  ALT=\"highlight%28side=about14.54cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >(rounded)<BR>\n" );
document.write( " <BR>\n" );
document.write( "USING TRIGONOMETRY:<BR>\n" );
document.write( "We can put one of those triangles on a set of coordinates to find the distance between the vertices of the octagon.<BR>\n" );
document.write( "Here are two similar triangles:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-0.1%2C1.2%2C-0.1%2C1.2%2C%0D%0Atriangle%280%2C0%2C1%2C0%2C0.707%2C0.707%29%2C%0D%0Ared%28line%28-0.1%2C0%2C1.2%2C0%29%29%2C%0D%0Ared%28line%280%2C-0.1%2C0%2C1.2%29%29%2C%0D%0Ared%28arrow%280.6%2C0%2C1.2%2C0%29%29%2C%0D%0Ared%28arrow%280%2C0.6%2C0%2C1.2%29%29%2C%0D%0Ared%28circle%280%2C0%2C1%29%29%2C%0D%0Ared%28arc%280%2C0%2C0.4%2C0.4%2C-45%2C0%29%29%2C%0D%0Alocate%280.2%2C0.15%2Cred%2845%5Eo%29%29%2C%0D%0Alocate%280.95%2C0%2CA%281%2C0%29%29%2Clocate%280.69%2C0.77%2CB%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28cos%2845%5Eo%29%2Csin%2845%5Eo%29%29 ALIGN=MIDDLE  ALT=\"B%28cos%2845%5Eo%29%2Csin%2845%5Eo%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28sqrt%282%29%2F2%2Csqrt%282%29%2F2%29 ALIGN=MIDDLE  ALT=\"B%28sqrt%282%29%2F2%2Csqrt%282%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%281%2C0%29 ALIGN=MIDDLE  ALT=\"A%281%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=sqrt%28%281-sqrt%282%29%2F2%29%5E2%2B%28sqrt%282%29%2F2%29%5E2%29=sqrt%281-sqrt%282%29%2B1%2F2%2B1%2F2%29=sqrt%282-sqrt%282%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-0.1%2C1.2%2C-0.1%2C1.2%2C%0D%0Atriangle%280%2C0%2C1%2C0%2C0.707%2C0.707%29%2C%0D%0Ared%28line%28-0.1%2C0%2C1.2%2C0%29%29%2C%0D%0Ared%28line%280%2C-0.1%2C0%2C1.2%29%29%2C%0D%0Ared%28arrow%280.6%2C0%2C1.2%2C0%29%29%2C%0D%0Ared%28arrow%280%2C0.6%2C0%2C1.2%29%29%2C%0D%0Ared%28arc%280%2C0%2C0.4%2C0.4%2C-45%2C0%29%29%2C%0D%0Alocate%280.2%2C0.15%2Cred%2845%5Eo%29%29%2C%0D%0Alocate%280.95%2C0%2CC%2819%2C0%29%29%2Clocate%280.69%2C0.77%2CD%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CD=highlight%2819sqrt%282-sqrt%282%29%29%29 ALIGN=MIDDLE  ALT=\"CD=highlight%2819sqrt%282-sqrt%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the length of the side of the octagon, in cm.<BR>\n" );
document.write( " <BR>\n" );
document.write( "USING LAW OF COSINES:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-0.1%2C1.2%2C-0.1%2C1.2%2C%0D%0Atriangle%280%2C0%2C1%2C0%2C0.707%2C0.707%29%2C%0D%0Ared%28arc%280%2C0%2C0.4%2C0.4%2C-45%2C0%29%29%2C%0D%0Alocate%280.2%2C0.15%2Cred%2845%5Eo%29%29%2C%0D%0Alocate%28-0.05%2C0%2CA%29%2Clocate%280.35%2C0.38%2Cc=19cm%29%2C%0D%0Alocate%280.4%2C0%2Cb=19cm%29%2Clocate%280.87%2C0.38%2Ca%29%2C%0D%0Alocate%281%2C0%2CC%29%2Clocate%280.69%2C0.77%2CB%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Law of cosine says that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=b%5E2%2Bc%5E2-2bc%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"a%5E2=b%5E2%2Bc%5E2-2bc%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so with length measured in cm,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=19%5E2%2B19%5E2-2%2A19%2A19%2Acos%2845%5Eo%29 ALIGN=MIDDLE  ALT=\"a%5E2=19%5E2%2B19%5E2-2%2A19%2A19%2Acos%2845%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=2%2A19%5E2-2%2A19%5E2%2A%28sqrt%282%29%2F2%29 ALIGN=MIDDLE  ALT=\"a%5E2=2%2A19%5E2-2%2A19%5E2%2A%28sqrt%282%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=2%2A19%5E2-19%5E2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"a%5E2=2%2A19%5E2-19%5E2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=19%5E2%282-sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"a%5E2=19%5E2%282-sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=sqrt%2819%5E2%282-sqrt%282%29%29%29 ALIGN=MIDDLE  ALT=\"a=sqrt%2819%5E2%282-sqrt%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=highlight%2819sqrt%282-sqrt%282%29%29%29 ALIGN=MIDDLE  ALT=\"a=highlight%2819sqrt%282-sqrt%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the length of the side of the octagon, in cm. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );