document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189710>Question 1189710</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the diagram attached below, Circles M and N are tangent to each other, and to Line AB and Line BC. If Angle ABC=120°, what is the ratio of the radius of Circle M to the radius of circle N\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Diagram: https://imgur.com/a/4dfUycL </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #821185 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=821185\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let r and R be the radii of circles N and M, respectively; the problem asks us to find R/r.<br><BR>\n" );
document.write( "Draw segment BM, intersecting circle M at D.<br><BR>\n" );
document.write( "Segment BM bisects angle ABC, so angles ABM and CBM are each 60 degrees.<br><BR>\n" );
document.write( "Draw segment NE with E the point of tangency of circle N to line BC.  Draw segment MF with F the point of tangency of circle M to line BC.<br><BR>\n" );
document.write( "Triangles BEN and BFM are similar 30-60-90 right triangles.<br><BR>\n" );
document.write( "The ratio of the hypotenuse to the long leg in a 30-60-90 right triangle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"2%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "In triangle BFM, FM is the long leg; the hypotenuse is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=MD%2BDN%2BNB=R%2Br%2Br%282%2Fsqrt%283%29%29 ALIGN=MIDDLE  ALT=\"MD%2BDN%2BNB=R%2Br%2Br%282%2Fsqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So in triangle BFM,<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28R%2Br%2Br%282%2Fsqrt%283%29%29%29%2FR=2%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"%28R%2Br%2Br%282%2Fsqrt%283%29%29%29%2FR=2%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Solve for R/r:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B%28r%2Br%282%2Fsqrt%283%29%29%29%2FR=2%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"1%2B%28r%2Br%282%2Fsqrt%283%29%29%29%2FR=2%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%281%2B2%2Fsqrt%283%29%29%29%2FR=2%2Fsqrt%283%29-1=%282-sqrt%283%29%29%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"%28r%281%2B2%2Fsqrt%283%29%29%29%2FR=2%2Fsqrt%283%29-1=%282-sqrt%283%29%29%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The current equation involves r/R on the left; take reciprocals to get an equation involving R/r:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%2F%28r%281%2B2%2Fsqrt%283%29%29%29=sqrt%283%29%2F%282-sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"R%2F%28r%281%2B2%2Fsqrt%283%29%29%29=sqrt%283%29%2F%282-sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%2Fr=%28sqrt%283%29%2F%282-sqrt%283%29%29%29%281%2B2%2Fsqrt%283%29%29 ALIGN=MIDDLE  ALT=\"R%2Fr=%28sqrt%283%29%2F%282-sqrt%283%29%29%29%281%2B2%2Fsqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%2Fr=%28sqrt%283%29%2F%282-sqrt%283%29%29%29%28%282%2Bsqrt%283%29%29%2Fsqrt%283%29%29 ALIGN=MIDDLE  ALT=\"R%2Fr=%28sqrt%283%29%2F%282-sqrt%283%29%29%29%28%282%2Bsqrt%283%29%29%2Fsqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%2Fr=%282%2Bsqrt%283%29%29%2F%282-sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"R%2Fr=%282%2Bsqrt%283%29%29%2F%282-sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%2Fr=%28%282%2Bsqrt%283%29%29%282%2Bsqrt%283%29%29%29%2F%28%282%2Bsqrt%283%29%29%282-sqrt%283%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%2Fr=%287%2B4sqrt%283%29%29%2F%284-3%29 ALIGN=MIDDLE  ALT=\"R%2Fr=%287%2B4sqrt%283%29%29%2F%284-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%2Fr+=+7%2B4sqrt%283%29 ALIGN=MIDDLE  ALT=\"R%2Fr+=+7%2B4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: The ratio of the radius of circle M to the radius of circle N is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%2Fr+=+7%2B4sqrt%283%29 ALIGN=MIDDLE  ALT=\"R%2Fr+=+7%2B4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );