document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=149173>Question 149173</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 10. The segments GA and GB are tangent to a circle with center O at A and B, and AGB is a 60-degree angle. Given that GA = 12 square root 3 cm, find the distance GO. Find the distance from G to the nearest point on the circle.<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 #109434 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=109434\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "First let's draw the picture (take note that I'm cutting AGB in half to make 2 30 degree angles):\r<BR>\n" );
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document.write( "From the drawing, we can see that segments GA and AO are legs of a triangle while GO is the hypotenuse. Since the angle that we are working with is the 30 degree angle, this means that GA is the adjacent side. So to find the hypotenuse, we need to use the cosine function:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29=adjacent%2Fhypotenuse ALIGN=MIDDLE  ALT=\"cos%28x%29=adjacent%2Fhypotenuse\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the cosine function.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2830%29=12%2Asqrt%283%29%2FGO ALIGN=MIDDLE  ALT=\"cos%2830%29=12%2Asqrt%283%29%2FGO\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the given sides and angle.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=GO%2Acos%2830%29=12%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"GO%2Acos%2830%29=12%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=GO ALIGN=MIDDLE  ALT=\"GO\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=GO=%2812%2Asqrt%283%29%29%2Fcos%2830%29 ALIGN=MIDDLE  ALT=\"GO=%2812%2Asqrt%283%29%29%2Fcos%2830%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2830%29 ALIGN=MIDDLE  ALT=\"cos%2830%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=GO=%2812%2Asqrt%283%29%29%2F%28sqrt%283%29%2F2%29 ALIGN=MIDDLE  ALT=\"GO=%2812%2Asqrt%283%29%29%2F%28sqrt%283%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2830%29 ALIGN=MIDDLE  ALT=\"cos%2830%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=GO=%2812%2Asqrt%283%29%29%2A%282%2Fsqrt%283%29%29 ALIGN=MIDDLE  ALT=\"GO=%2812%2Asqrt%283%29%29%2A%282%2Fsqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply by the reciprocal\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=GO=%2812%2Across%28sqrt%283%29%29%29%2A%282%2Fcross%28sqrt%283%29%29%29 ALIGN=MIDDLE  ALT=\"GO=%2812%2Across%28sqrt%283%29%29%29%2A%282%2Fcross%28sqrt%283%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cancel like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=GO=12%2A2 ALIGN=MIDDLE  ALT=\"GO=12%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=GO=24 ALIGN=MIDDLE  ALT=\"GO=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "So the length of GO is 24 units.\r<BR>\n" );
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document.write( "The closest point to G will lie on the line GO. So to find the distance from G to this point, we need to find the radius. \r<BR>\n" );
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document.write( "Using the previous drawing, we can see that the radius is AO. To find the length of AO, note that AO is the opposite leg to the angle 30 degrees. So let's use the sine function to find AO\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29=opposite%2Fhypotenuse ALIGN=MIDDLE  ALT=\"sin%28x%29=opposite%2Fhypotenuse\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the sine function.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%2830%29=AO%2F24 ALIGN=MIDDLE  ALT=\"sin%2830%29=AO%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the angle 30 , the opposite side AO, and the hypotenuse 24\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2=AO%2F24 ALIGN=MIDDLE  ALT=\"1%2F2=AO%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the sine of 30 to get <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24=2AO ALIGN=MIDDLE  ALT=\"24=2AO\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cross multiply.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12=AO ALIGN=MIDDLE  ALT=\"12=AO\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 2.\r<BR>\n" );
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document.write( "So the length of AO (the radius) is 12 units.\r<BR>\n" );
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document.write( "Now simply subtract the radius from the length GO to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24-12=12 ALIGN=MIDDLE  ALT=\"24-12=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the distance from G to the nearest point on the circle is 12 units. </SPAN>\n" );
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