document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1074278>Question 1074278</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In triangle RST, U lies on TS with TU:US= 2:3. M is the midpoint of RU. TM intersects RS in V. Find the ratio of RV:RS. <BR>\n" );
document.write( "I was able to solve it with trig, and I got 2:7. It was meant to be solved with basic geometry, but I haven't be able to think of a way.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #689017 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=689017\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Maybe by now you have been able to think of a way.<BR>\n" );
document.write( "I could not think of a way late last night,<BR>\n" );
document.write( "but I woke up with a clearer mind this morning,<BR>\n" );
document.write( "so I added line MW, parallel to ST:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1%2C11%2C-1%2C11%2C%0D%0Agreen%28triangle%281%2C5%2C3%2C5%2C1.25%2C6.25%29%29%2C%0D%0Atriangle%281.25%2C6.25%2C10%2C0%2C2%2C10%29%2C%0D%0Atriangle%280%2C0%2C4%2C0%2C2%2C10%29%2C%0D%0Aline%280%2C0%2C10%2C0%29%2Cline%280%2C0%2C2%2C10%29%2C%0D%0Aline%282%2C10%2C10%2C0%29%2Clocate%28-0.2%2C0%2CS%29%2C%0D%0Alocate%283.9%2C0%2CU%29%2Clocate%289.9%2C0%2CT%29%2C%0D%0Alocate%281.9%2C10.6%2CR%29%2Clocate%283.1%2C5.5%2CM%29%2C%0D%0Alocate%280.9%2C6.8%2CV%29%2Clocate%280.5%2C5.3%2CW%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > MW is part of two new triangles similar to two triangles that were there before.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Now that I look at the drawing,<BR>\n" );
document.write( "I realize it has TU:RS = 3:2 .<BR>\n" );
document.write( "A drawing is not necessary to support the reasoning,<BR>\n" );
document.write( "but even an inaccurate drawing helps<BR>\n" );
document.write( "as a reminder of the meaning of each point.<BR>\n" );
document.write( " <BR>\n" );
document.write( "RWM is similar to RSU.<BR>\n" );
document.write( "MVW is similar to TVS.<BR>\n" );
document.write( "The \"scale down factors\" between the similar triangles<BR>\n" );
document.write( "lead to a way to calculate the lengths of RW and VW<BR>\n" );
document.write( "as fractions of the length pf RS,<BR>\n" );
document.write( "and that allows to calculate, as a fraction of the length of RS,<BR>\n" );
document.write( "the lengths of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=RV=RW-VW ALIGN=MIDDLE  ALT=\"RV=RW-VW\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=TU%2BUS=TS ALIGN=MIDDLE  ALT=\"TU%2BUS=TS\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=TU%3AUS+=+2%3A3 ALIGN=MIDDLE  ALT=\"TU%3AUS+=+2%3A3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=TU%3AUS%3ATS+=+2%3A3%3A3%2B2=5 ALIGN=MIDDLE  ALT=\"TU%3AUS%3ATS+=+2%3A3%3A3%2B2=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=TU=%282%2F5%29TS ALIGN=MIDDLE  ALT=\"TU=%282%2F5%29TS\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=US=%283%2F5%29TS ALIGN=MIDDLE  ALT=\"US=%283%2F5%29TS\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since M is the midpoint of RU,<BR>\n" );
document.write( "and WM is parallel to ST,<BR>\n" );
document.write( "RWM is not only similar to RSU;<BR>\n" );
document.write( "it is a <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;\" > scale version of RSU.<BR>\n" );
document.write( "So, not only is MW the midsegment of RSU,<BR>\n" );
document.write( "with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=MW=%281%2F2%29US ALIGN=MIDDLE  ALT=\"MW=%281%2F2%29US\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but also <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=RW=%281%2F2%29RS ALIGN=MIDDLE  ALT=\"RW=%281%2F2%29RS\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and of course, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=WS=%281%2F2%29RS ALIGN=MIDDLE  ALT=\"WS=%281%2F2%29RS\"   DISABLED_style= \"max-width:90%; height:auto;\" > too.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=MW=%281%2F2%29US=%281%2F2%29%283%2F5%29TS=%283%2F10%29TS ALIGN=MIDDLE  ALT=\"MW=%281%2F2%29US=%281%2F2%29%283%2F5%29TS=%283%2F10%29TS\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since MVW is similar to TVS,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=MW=%283%2F10%29TS ALIGN=MIDDLE  ALT=\"MW=%283%2F10%29TS\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=VW=%283%2F10%29VS ALIGN=MIDDLE  ALT=\"VW=%283%2F10%29VS\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Finally,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28VS=VW%2BWS=VW%2B%281%2F2%29RS%2CVW=%283%2F10%29VS%29 ALIGN=MIDDLE  ALT=\"system%28VS=VW%2BWS=VW%2B%281%2F2%29RS%2CVW=%283%2F10%29VS%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=VS=%283%2F10%29VS%2B%281%2F2%29RS ALIGN=MIDDLE  ALT=\"VS=%283%2F10%29VS%2B%281%2F2%29RS\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Form there,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=VS-%283%2F10%29VS=%281%2F2%29RS ALIGN=MIDDLE  ALT=\"VS-%283%2F10%29VS=%281%2F2%29RS\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%2F10%29VS=%281%2F2%29RS ALIGN=MIDDLE  ALT=\"%287%2F10%29VS=%281%2F2%29RS\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=VS=%2810%2F7%29%281%2F2%29RS ALIGN=MIDDLE  ALT=\"VS=%2810%2F7%29%281%2F2%29RS\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=VS=%285%2F7%29RS ALIGN=MIDDLE  ALT=\"VS=%285%2F7%29RS\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=RV=RS-VS ALIGN=MIDDLE  ALT=\"RV=RS-VS\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=RV=RS-%285%2F7%29RS ALIGN=MIDDLE  ALT=\"RV=RS-%285%2F7%29RS\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28RV=%282%2F7%29RS%29 ALIGN=MIDDLE  ALT=\"highlight%28RV=%282%2F7%29RS%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "I am not sure if the reasoning above is the shortest way to the answer.<BR>\n" );
document.write( "Maybe I should have drawn a line through M parallel to RS instead.<BR>\n" );
document.write( "With nothing else to do, I would explore other angles to this problem<BR>\n" );
document.write( "(and draw a better drawing).<BR>\n" );
document.write( "However, if you are still looking for a geometry-based answer,<BR>\n" );
document.write( "a less than perfect answer is better than no answer. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );