document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=550587>Question 550587</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.M is the mid-point of a line segment AB;AXB and MYB are equilateral triangles on opposite sides of AB;XY cuts AB at Z.Prove that:AZ=2 ZB.<BR>\n" );
document.write( "2.In trapezium ABCD,AB//DC and DC=2 AB.EF,drawn parallel to AB cuts AD in F and BC in E such that 4 BE=3 EC.Diagonal DB intersects FE at point G.Prove that:7 EF=10 AB.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #359079 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=359079\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28200%2C300%2C-2.4%2C2.4%2C-2.6%2C4.6%2C%0D%0Atriangle%28-2%2C0%2C2%2C0%2C0%2C3.464%29%2C%0D%0Atriangle%280%2C0%2C2%2C0%2C1%2C-1.732%29%2C%0D%0Ared%28line%280%2C3.464%2C1%2C-1.732%29%29%2C%0D%0Alocate%28-0.1%2C3.8%2CX%29%2C+locate%280.9%2C-1.7%2CY%29%2C%0D%0Alocate%28-2.1%2C0%2CA%29%2C+locate%282.1%2C0.3%2CB%29%2C%0D%0Alocate%28-0.15%2C0%2CM%29%2C+locate%280.65%2C0.4%2CZ%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Since M is the midpoint of AB, AM=MB and AB=2AM=2MB<BR>\n" );
document.write( "Since ABX and ABY are equilateral, AB=BX=AX=2MB=2BY=2MY<BR>\n" );
document.write( "Triangles BXZ and BZY have congruent vertical angles at Z, and congruent 60 degree angles ZBX and ZMY. They are similar, and since BX=2BY, ZB=2MZ.<BR>\n" );
document.write( "Then AM=MB=MZ+ZB=3MZ.<BR>\n" );
document.write( "So AZ=AM+MZ=3MZ+MZ=4MZ,<BR>\n" );
document.write( "and since ZB=2MZ, AZ/ZB=2MZ/2MZ=2<BR>\n" );
document.write( "AZ/ZB=2 --> AZ=2ZB\r<BR>\n" );
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document.write( "2. I used to call ABCD a trapezium too, but I'm going to call it a trapezoid, because I live in the USA now.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1%2C9%2C-1%2C9%2C%0D%0Agrid%281%29%2C%0D%0Ablue%28line%280%2C8%2C8%2C8%29%29%2C+blue%28line%281%2C1%2C5%2C1%29%29%2C%0D%0Ablue%28line%280%2C8%2C1%2C1%29%29%2C+blue%28line%288%2C8%2C5%2C1%29%29%2C%0D%0Agreen%28line%280%2C8%2C5%2C1%29%29%2C+red%28line%284%2F7%2C4%2C6%262%2F7%2C4%29%29%2C%0D%0Alocate%280.7%2C1%2CA%29%2Clocate%285.1%2C1%2CB%29%2Clocate%288.2%2C8.5%2CC%29%2Clocate%280.2%2C8.5%2CD%29%2C%0D%0Alocate%280.2%2C4%2CF%29%2Clocate%283%2C4.5%2CG%29%2Clocate%286.5%2C4%2CE%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The red line FE divides each of the 3 slanted lines into two segments with lengths in the ratio 4:3.<BR>\n" );
document.write( "On line BC<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+BE=3+EC ALIGN=MIDDLE  ALT=\"4+BE=3+EC\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BE=%283%2F4%29EC ALIGN=MIDDLE  ALT=\"BE=%283%2F4%29EC\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=BE%2BEC=%287%2F4%29EC ALIGN=MIDDLE  ALT=\"BC=BE%2BEC=%287%2F4%29EC\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=EC=%284%2F7%29BC ALIGN=MIDDLE  ALT=\"EC=%284%2F7%29BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BE=%283%2F7%29BC ALIGN=MIDDLE  ALT=\"BE=%283%2F7%29BC\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Similar ratios can be written for the segments on lines DB and AD.<BR>\n" );
document.write( "Diagonal DB divides the trapezoid into 2 triangles (BCD and ABD). Each of those triangles has a smaller similar triangle to one side of red line FE (BEG and FGD respectively).<BR>\n" );
document.write( "The ratio of the side lengths of BEG and BCD is 3:7, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=GE=%283%2F7%29DC ALIGN=MIDDLE  ALT=\"GE=%283%2F7%29DC\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The ratio of the side lengths of FGD and ABD is 4:7, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=FG=%284%2F7%29AB ALIGN=MIDDLE  ALT=\"FG=%284%2F7%29AB\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Since DC=2 AB,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=FE=FG%2BGE=%284%2F7%29AB%2B%283%2F7%29DC=%284%2F7%29AB%2B%283%2F7%29%282%2AAB%29=%284%2F7%29AB%2B%286%2F7%29AB=%2810%2F7%29AB ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=FE=%2810%2F7%29AB ALIGN=MIDDLE  ALT=\"FE=%2810%2F7%29AB\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7FE=10AB ALIGN=MIDDLE  ALT=\"7FE=10AB\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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