document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=865766>Question 865766</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, could you please help me with the question (10ii) in the following exam paper. This is the link, thank you. \r<BR>\n" );
document.write( "\n" );
document.write( "http://papers.xtremepapers.com/CIE/Cambridge%20International%20A%20and%20AS%20Level/Mathematics%20(9709)/9709_s10_qp_11.pdf </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #521901 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=521901\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> NOTE: Check my calculations, because it is way past my bedtime.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The diagonals are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%2820%2C20%2C-0.2%2C1.2%2C-0.2%2C1.2%2Carrow%28-0.1%2C1%2C1.1%2C1%29%2Clocate%280%2C0.9%2COB%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%2820%2C20%2C-0.2%2C1.2%2C-0.2%2C1.2%2Carrow%28-0.1%2C1%2C1.1%2C1%29%2Clocate%280%2C0.9%2COA%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B%22 ALIGN=MIDDLE  ALT=\"%22%2B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%2820%2C20%2C-0.2%2C1.2%2C-0.2%2C1.2%2Carrow%28-0.1%2C1%2C1.1%2C1%29%2Clocate%280%2C0.9%2COC%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28i%2B3j%2B3k%29%2B%283i-j%2Bk%29=4i%2B2j%2B4k ALIGN=MIDDLE  ALT=\"%28i%2B3j%2B3k%29%2B%283i-j%2Bk%29=4i%2B2j%2B4k\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "AND<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%2820%2C20%2C-0.2%2C1.2%2C-0.2%2C1.2%2Carrow%28-0.1%2C1%2C1.1%2C1%29%2Clocate%280%2C0.9%2CAC%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%2820%2C20%2C-0.2%2C1.2%2C-0.2%2C1.2%2Carrow%28-0.1%2C1%2C1.1%2C1%29%2Clocate%280%2C0.9%2COC%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22-%22 ALIGN=MIDDLE  ALT=\"%22-%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%2820%2C20%2C-0.2%2C1.2%2C-0.2%2C1.2%2Carrow%28-0.1%2C1%2C1.1%2C1%29%2Clocate%280%2C0.9%2COA%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283i-j%2Bk%29-%28i%2B3j%2B3k%29=2i-4j-2k%29 ALIGN=MIDDLE  ALT=\"%283i-j%2Bk%29-%28i%2B3j%2B3k%29=2i-4j-2k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OB=sqrt%284%5E2%2B2%5E2%2B4%5E2%29=sqrt%2816%2B4%2B16%29=sqrt%2836%29=6 ALIGN=MIDDLE  ALT=\"OB=sqrt%284%5E2%2B2%5E2%2B4%5E2%29=sqrt%2816%2B4%2B16%29=sqrt%2836%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC=sqrt%282%5E2%2B4%5E2%2B2%5E2%29=sqrt%284%2B16%2B4%29=sqrt%2824%29=2sqrt%286%29 ALIGN=MIDDLE  ALT=\"AC=sqrt%282%5E2%2B4%5E2%2B2%5E2%29=sqrt%284%2B16%2B4%29=sqrt%2824%29=2sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The diagonals bisect each other and split parallelogram OABC into 4 triangles.<BR>\n" );
document.write( "Two sides of each triangle will be the halves of the two diagonals.<BR>\n" );
document.write( "That is, two sides will have lengths of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F2=3 ALIGN=MIDDLE  ALT=\"6%2F2=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%286%29%2F2=sqrt%286%29 ALIGN=MIDDLE  ALT=\"2sqrt%286%29%2F2=sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The remaining side of each triangle will be one side of the parallelogram, and not all sides of the parallelogram are the same length. (It is not a rhombus).<BR>\n" );
document.write( "The parallelogram's side lengths are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OA=sqrt%281%5E2%2B3%5E3%2B3%5E2%29=sqrt%281%2B9%2B9%29=sqrt%2819%29 ALIGN=MIDDLE  ALT=\"OA=sqrt%281%5E2%2B3%5E3%2B3%5E2%29=sqrt%281%2B9%2B9%29=sqrt%2819%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OC=sqrt%283%5E3%2B1%5E2%2B1%5E2%29=sqrt%289%2B1%2B1%29=sqrt%2811%29 ALIGN=MIDDLE  ALT=\"OC=sqrt%283%5E3%2B1%5E2%2B1%5E2%29=sqrt%289%2B1%2B1%29=sqrt%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The angle between the sides measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%286%29 ALIGN=MIDDLE  ALT=\"sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be smaller in the two triangles where it is opposite the shorter parallelogram side measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2811%29 ALIGN=MIDDLE  ALT=\"sqrt%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We need the measure of that angle, that I will call <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The law of cosines, applied to that triangle says that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%2811%29%29%5E2=%283%29%5E2%2B%28sqrt%286%29%29%5E2%29-2%283%29%28sqrt%286%29%29cos%28theta%29 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=11=9%2B6-6sqrt%286%29cos%28theta%29 ALIGN=MIDDLE  ALT=\"11=9%2B6-6sqrt%286%29cos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6sqrt%286%29cos%28theta%29=9%2B6-11 ALIGN=MIDDLE  ALT=\"6sqrt%286%29cos%28theta%29=9%2B6-11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6sqrt%286%29cos%28theta%29=4 ALIGN=MIDDLE  ALT=\"6sqrt%286%29cos%28theta%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29=4%2F6sqrt%286%29 ALIGN=MIDDLE  ALT=\"cos%28theta%29=4%2F6sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29=2%2F3sqrt%286%29 ALIGN=MIDDLE  ALT=\"cos%28theta%29=2%2F3sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29=2sqrt%286%29%2F%283%2A6%29 ALIGN=MIDDLE  ALT=\"cos%28theta%29=2sqrt%286%29%2F%283%2A6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29=sqrt%286%29%2F9 ALIGN=MIDDLE  ALT=\"cos%28theta%29=sqrt%286%29%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=74.2%5Eo ALIGN=MIDDLE  ALT=\"theta=74.2%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );