document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1052636>Question 1052636</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if an equilateral triangle is circumscribed about a circle of radius 10cm. determine the side of the triangle?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #667955 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=667955\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The circle and equilateral triangle can be slices pizza-style into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > wedges/triangles.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C330%2C-2%2C2%2C-1.3%2C2%2C%0D%0Acircle%280%2C0%2C1%29%2Ccircle%280%2C0%2C0.04%29%2C%0D%0Atriangle%28-1.79%2C-1%2C1.79%2C-1%2C0%2C1.79%29%2C%0D%0Atriangle%28-1.79%2C-1%2C0%2C-1%2C0%2C0%29%2C%0D%0Arectangle%280%2C-1%2C-0.1%2C-0.9%29%2C%0D%0Alocate%28-0.3%2C-0.3%2Cred%2860%5Eo%29%29%2C%0D%0Ared%28arc%280%2C0%2C0.7%2C0.7%2C90%2C150%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The right triangle shown above is one of those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > triangles.<BR>\n" );
document.write( "It is a right triangle because the radius of the circle must be perpendicular to the tangent side of the equilateral triangle.<BR>\n" );
document.write( "The angle at the center of the circle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo%2F6=red%2860%5Eo%29 ALIGN=MIDDLE  ALT=\"360%5Eo%2F6=red%2860%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The side adjacent to that angle is the radius of the circle, and measures <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10cm ALIGN=MIDDLE  ALT=\"10cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The opposite side, of length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > , is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the side of the equilateral triangle.<BR>\n" );
document.write( "The trigonometric ratios tell us that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F%2210+cm%22=tan%2860%5Eo%29=sqrt%283%29=about1.73 ALIGN=MIDDLE  ALT=\"x%2F%2210+cm%22=tan%2860%5Eo%29=sqrt%283%29=about1.73\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=10sqrt%283%29 ALIGN=MIDDLE  ALT=\"x=10sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm=about17.3cm ALIGN=MIDDLE  ALT=\"cm=about17.3cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x=highlight%2820sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"2x=highlight%2820sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm=highlight%28about34.6cm%29 ALIGN=MIDDLE  ALT=\"cm=highlight%28about34.6cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the length of the side of the equilateral triangle.<BR>\n" );
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document.write( "Allergic to trigonometry? Then, you would have to use similar triangles and the Pythagorean theorem.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C330%2C-2%2C2%2C-1.3%2C2%2C%0D%0Acircle%280%2C0%2C1%29%2Ccircle%280%2C0%2C0.04%29%2C%0D%0Atriangle%28-1.79%2C-1%2C1.79%2C-1%2C0%2C1.79%29%2C%0D%0Atriangle%28-1.79%2C-1%2C0%2C-1%2C0%2C0%29%2C%0D%0Atriangle%281.79%2C-1%2C0%2C1.79%2C0%2C-1%29%2C%0D%0Arectangle%280%2C-1%2C-0.1%2C-0.9%29%2C%0D%0Alocate%28-1.85%2C-1%2CA%29%2Clocate%281.78%2C-1%2CB%29%2C%0D%0Alocate%28-0.03%2C1.95%2CC%29%2Clocate%28-0.02%2C-1%2CD%29%2C%0D%0Alocate%280.05%2C0.03%2CO%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Right triangles BCD and OAD are similar because their smallest angles are congruent. (Angles BCD and OAD are both half of an equilateral triangle's angle).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD=DC=x ALIGN=MIDDLE  ALT=\"AD=DC=x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is half of the equilateral triangle's side.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC=BC=2x ALIGN=MIDDLE  ALT=\"AC=BC=2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the equilateral triangle's side.<BR>\n" );
document.write( "Since the triangles are similar, the ratio of long leg to short leg is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD%2FDO=CD%2FBC ALIGN=MIDDLE  ALT=\"AD%2FDO=CD%2FBC\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F%2210+cm%22=CD%2Fx ALIGN=MIDDLE  ALT=\"x%2F%2210+cm%22=CD%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=CD%2A10cm ALIGN=MIDDLE  ALT=\"x%5E2=CD%2A10cm\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can find CD, the long leg of BCD, using the Pythagorean theorem, because<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CD%5E2%2BDC%5E2=BC%5E2 ALIGN=MIDDLE  ALT=\"CD%5E2%2BDC%5E2=BC%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CD%5E2%2Bx%5E2=%282x%29%5E2 ALIGN=MIDDLE  ALT=\"CD%5E2%2Bx%5E2=%282x%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CD%5E2%2Bx%5E2=%282x%29%5E2 ALIGN=MIDDLE  ALT=\"CD%5E2%2Bx%5E2=%282x%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CD%5E2%2Bx%5E2=4x%5E2 ALIGN=MIDDLE  ALT=\"CD%5E2%2Bx%5E2=4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CD%5E2=3x%5E2 ALIGN=MIDDLE  ALT=\"CD%5E2=3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CD=sqrt%283%29x ALIGN=MIDDLE  ALT=\"CD=sqrt%283%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, plugging that into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=CD%2A10cm ALIGN=MIDDLE  ALT=\"x%5E2=CD%2A10cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=sqrt%283%29x%2A10cm ALIGN=MIDDLE  ALT=\"x%5E2=sqrt%283%29x%2A10cm\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%283%29%2A10cm ALIGN=MIDDLE  ALT=\"x=sqrt%283%29%2A10cm\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the side of the equilateral triangle measures <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC=2x=2sqrt%283%29%2A10cm=highlight%28sqrt%283%29%2A20cm=about+34.6cm%29 ALIGN=MIDDLE  ALT=\"AC=2x=2sqrt%283%29%2A10cm=highlight%28sqrt%283%29%2A20cm=about+34.6cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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