document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=642714>Question 642714</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that Parallelogram ABCD has vertices A(2, 4), B(8, 4), C(5, -3) and D(-1, -3). What is the perimeter of the Parallelogram ABCD? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #404208 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=404208\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Sides AB and CD are horizontal (constant y).<BR>\n" );
document.write( "They are supposed to be parallel and congruent (equal length) because opposite sides in a parallelogram are parallel and congruent. So the length of ab and CD are the same (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=CD ALIGN=MIDDLE  ALT=\"AB=CD\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and I only need to calculate one of those distances.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-2%2C10%2C-6%2C6%2C%0D%0Agrid%281%29%2C%0D%0Ared%28circle%282%2C4%2C0.2%29%29%2Cred%28circle%288%2C4%2C0.2%29%29%2C%0D%0Ared%28circle%285%2C-3%2C0.2%29%29%2Cred%28circle%28-1%2C-3%2C0.2%29%29%2C%0D%0Alocate%281.5%2C4.5%2CA%29%2Clocate%288.2%2C4.5%2CB%29%2C%0D%0Alocate%285.2%2C-3%2CC%29%2Clocate%28-1.5%2C-3%2CD%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The diatance from A to B (the length of AB) is the positive difference in x coordinate between the ends:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=abs%288-2%29=6 ALIGN=MIDDLE  ALT=\"AB=abs%288-2%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The other two sides (BC and AD) are also parallel and equal length (because it is a parallelogram). So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=AD ALIGN=MIDDLE  ALT=\"BC=AD\"   DISABLED_style= \"max-width:90%; height:auto;\" > and I only need to calculaye one of those lengths.<BR>\n" );
document.write( "The length of BC is the distance from B to C.<BR>\n" );
document.write( "It is calculated using a formula that is derived from the Pythagorean theorem.<BR>\n" );
document.write( "You do not need to memorize formulas to know math or to please me.<BR>\n" );
document.write( "Your teacher may like to see formulas written first, with all the confusing symbols.<BR>\n" );
document.write( "You may have given a formula that looks like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=sqrt%28%28x%5BC%5D-x%5BB%5D%29%5E2%2B%28y%5BC%5D-y%5BB%5D%29%5E2%29 ALIGN=MIDDLE  ALT=\"BC=sqrt%28%28x%5BC%5D-x%5BB%5D%29%5E2%2B%28y%5BC%5D-y%5BB%5D%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The problem is that different people use different symbols.<BR>\n" );
document.write( "If you must write the formula to please your teacher, write it just like it was done in class.<BR>\n" );
document.write( "In any case, the symbol to the left of the equal sign (AB, or d, or whatever) represents the distance between two points.<BR>\n" );
document.write( "What's in the brackets is the difference in x coordinates between the two points, and the difference of y coordinates between the two points.<BR>\n" );
document.write( "The difference in x coordinates between B and C is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BC%5D-x%5BB%5D=5-8 ALIGN=MIDDLE  ALT=\"x%5BC%5D-x%5BB%5D=5-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >. That goes in brackets (to be calculated first) and is squared.<BR>\n" );
document.write( "The difference in y coordinates between B and C is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5BC%5D-y%5BB%5D=-3-4 ALIGN=MIDDLE  ALT=\"y%5BC%5D-y%5BB%5D=-3-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. That, in brackets and squared, is added before taking the square root of the whole mess.<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=sqrt%28%285-8%29%5E2%2B%28-3-4%29%5E2%29=sqrt%28%28-3%29%5E2%2B%28-7%29%5E2%29=sqrt%289%2B49%29=sqrt%2858%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The perimeter of the parallelogram is the sum of all sides,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perimeter=AB%2BBC%2BCD%2CAD ALIGN=MIDDLE  ALT=\"perimeter=AB%2BBC%2BCD%2CAD\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Because opposite sides are congruent (equal lengths),<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=CD ALIGN=MIDDLE  ALT=\"AB=CD\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=AD ALIGN=MIDDLE  ALT=\"BC=AD\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perimeter=AB%2BBC%2BAB%2BBC ALIGN=MIDDLE  ALT=\"perimeter=AB%2BBC%2BAB%2BBC\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perimeter=%28AB%2BBC%29%2B%28AB%2BBC%29 ALIGN=MIDDLE  ALT=\"perimeter=%28AB%2BBC%29%2B%28AB%2BBC%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28perimeter=2%2A%28AB%2BBC%29%29 ALIGN=MIDDLE  ALT=\"highlight%28perimeter=2%2A%28AB%2BBC%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "That last formula, which some teachers may want you to write, with whatever simbols wer used in class, is what I used to calculate the perimeter, calculated as 2 times the sum of two non-parallel, sides (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=6 ALIGN=MIDDLE  ALT=\"AB=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=sqrt%2858%29 ALIGN=MIDDLE  ALT=\"BC=sqrt%2858%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perimeter=2%286%2Bsqrt%2858%29%29 ALIGN=MIDDLE  ALT=\"perimeter=2%286%2Bsqrt%2858%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28perimeter=12%2B2sqrt%2858%29%29 ALIGN=MIDDLE  ALT=\"highlight%28perimeter=12%2B2sqrt%2858%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );