document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165801>Question 1165801</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The quadrilateral PQRS has vertices P(2,4), Q(5,1), R(-1,-2) and S(-4,1). Prove that PQRS is a parallelogram by showing that the opposite sides are equal. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #790329 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=790329\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The quadrilateral<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+PQRS ALIGN=MIDDLE  ALT=\"+PQRS\"   DISABLED_style= \"max-width:90%; height:auto;\" > has vertices <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >) <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q ALIGN=MIDDLE  ALT=\"Q\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >) <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S ALIGN=MIDDLE  ALT=\"S\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( " \r<BR>\n" );
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document.write( " the opposite sides are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ ALIGN=MIDDLE  ALT=\"PQ\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=RS ALIGN=MIDDLE  ALT=\"RS\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PS ALIGN=MIDDLE  ALT=\"PS\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=QR ALIGN=MIDDLE  ALT=\"QR\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "use given data to find distance between points or the length of the sides:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >) <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q ALIGN=MIDDLE  ALT=\"Q\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=formula_distance.solver>Distance Between 2 points</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>The distance formula is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Plug in the numbers,\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28%285-%282%29%29%5E2%29%2B%28%281-%284%29%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28%285-%282%29%29%5E2%29%2B%28%281-%284%29%29%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%5E2%2B-3%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%283%5E2%2B-3%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  The distance is 4.24264068711928.\r<BR>\n" );
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document.write( "</TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ=4.24264068711928 ALIGN=MIDDLE  ALT=\"PQ=4.24264068711928\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S ALIGN=MIDDLE  ALT=\"S\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=formula_distance.solver>Distance Between 2 points</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>The distance formula is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Plug in the numbers,\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28%28-4-%28-1%29%29%5E2%29%2B%28%281-%28-2%29%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28%28-4-%28-1%29%29%5E2%29%2B%28%281-%28-2%29%29%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-3%5E2%2B3%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28-3%5E2%2B3%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  The distance is 4.24264068711928.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "</TD></TR>\n" );
document.write( "</TABLE>\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=RS=4.24264068711928 ALIGN=MIDDLE  ALT=\"RS=4.24264068711928\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "so, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ=RS ALIGN=MIDDLE  ALT=\"PQ=RS\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "if so, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PS ALIGN=MIDDLE  ALT=\"PS\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=QR ALIGN=MIDDLE  ALT=\"QR\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be equal too\r<BR>\n" );
document.write( "\n" );
document.write( "let's check it\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >) <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S ALIGN=MIDDLE  ALT=\"S\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=formula_distance.solver>Distance Between 2 points</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>The distance formula is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Plug in the numbers,\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28%28-4-%282%29%29%5E2%29%2B%28%281-%284%29%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28%28-4-%282%29%29%5E2%29%2B%28%281-%284%29%29%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-6%5E2%2B-3%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28-6%5E2%2B-3%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  The distance is 6.70820393249937.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "</TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PS=6.70820393249937 ALIGN=MIDDLE  ALT=\"PS=6.70820393249937\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q ALIGN=MIDDLE  ALT=\"Q\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) \r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=formula_distance.solver>Distance Between 2 points</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>The distance formula is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Plug in the numbers,\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28%28-1-%285%29%29%5E2%29%2B%28%28-2-%281%29%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28%28-1-%285%29%29%5E2%29%2B%28%28-2-%281%29%29%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-6%5E2%2B-3%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28-6%5E2%2B-3%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  The distance is 6.70820393249937.\r<BR>\n" );
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document.write( "</TABLE><BR>\n" );
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document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=QR=6.70820393249937 ALIGN=MIDDLE  ALT=\"QR=6.70820393249937\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PS+=+QR ALIGN=MIDDLE  ALT=\"PS+=+QR\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "it is proven that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQRS+ ALIGN=MIDDLE  ALT=\"PQRS+\"   DISABLED_style= \"max-width:90%; height:auto;\" >is a parallelogram  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%0D%0Acircle%285%2C1%2C.12%29%2Ccircle%28-1%2C-2%2C.12%29%2C%0D%0Acircle%282%2C4%2C.12%29%2Ccircle%28-4%2C1%2C.12%29%2C%0D%0Agreen%28line%285%2C1%2C-1%2C-2%29%29%2Cgreen%28line%282%2C4%2C-4%2C1%29%29%2C%0D%0Agreen%28line%28-1%2C-2%2C-4%2C1%29%29%2Cgreen%28line%282%2C4%2C5%2C1%29%29%2C%0D%0A+graph%28+600%2C+600%2C+-10%2C+10%2C-10%2C10%2C+0%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );