document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1062408>Question 1062408</A>: <I><SPAN STYLE=\"word-break: break-all;\"> P is the midpoint of segment BC in triangle ABC and Q is the midpoint of seg AP .ray BQ cuts segment AC at R. show that : BQ = 3QR </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #677479 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=677479\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here is a triangle ABC, including points P, Q, R, a few extra midpoints, and a midsegment.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-5%2C5%2C-1%2C9%2C%0D%0Atriangle+%28-4%2C0%2C4%2C0%2C2%2C8%29%2C%0D%0Atriangle+%28-4%2C0%2C0%2C5.333%2C4%2C0%29%2C%0D%0Atriangle+%280%2C0%2C4%2C0%2C2%2C8%29%2C%0D%0Agreen+%28line+%28-1%2C4%2C3%2C4%29%29%2C%0D%0Alocate%282%2C8.5%2CA%29%2Clocate%284%2C0%2CB%29%2Clocate%28-4.1%2C0%2CC%29%2C%0D%0Alocate+%28-0.1%2C0%2CP%29%2Clocate+%281.2%2C4.5%2CQ%29%2Clocate+%28-0.3%2C5.8%2CR%29%2C%0D%0Alocate+%28-1.4%2C4.25%2CS%29%2Clocate+%28-2.1%2C-0.1%2CU%29%2Clocate+%283.2%2C4.25%2CT%29%2C%0D%0Ared%28circle%28-2%2C0%2C0.05%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > S, T, and U are the midpoints of AC,  AB,  and PC respectively.<BR>\n" );
document.write( "Of course, ST is the midsegment of ABC, and SQ is the midsegment of APC.\r<BR>\n" );
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document.write( "Since midsegments are half a long as the base, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=SQ=%281%2F2%29PC=%281%2F4%29BC=PU=UC ALIGN=MIDDLE  ALT=\"SQ=%281%2F2%29PC=%281%2F4%29BC=PU=UC\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since SQ and UC are congruent and parallel,<BR>\n" );
document.write( "SQUC is a parallelogram,<BR>\n" );
document.write( "and QU is parallel to AC .<BR>\n" );
document.write( "With their pairs of parallel sides, triangles SQR and UBQ have 3 pairs of congruent angles.<BR>\n" );
document.write( "That makes triangles SQR and UBQ similar triangles.<BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=UC=%281%2F4%29BC ALIGN=MIDDLE  ALT=\"UC=%281%2F4%29BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=UB=%283%2F4%29BC ALIGN=MIDDLE  ALT=\"UB=%283%2F4%29BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The corresponding side in SQR is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=SQ=%281%2F4%29BC ALIGN=MIDDLE  ALT=\"SQ=%281%2F4%29BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "So, sides of UBQ are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > times longer than corresponding sides of SQR :<BR>\n" );
document.write( "UB = 3SQ , UQ = 3SR , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight+%28BQ=3QR%29 ALIGN=MIDDLE  ALT=\"highlight+%28BQ=3QR%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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document.write( " </SPAN>\n" );
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