document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208799>Question 1208799</A>: <I><SPAN STYLE=\"word-break: break-all;\"> ABCD is a quadrilateral. If diagonals AC and BD bisect each other, then prove that ABCD is a parallelogram. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847249 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847249\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "E = intersection of diagonals AC and BD<BR>\n" );
document.write( "By construction we'll have EA = EC and EB = ED be the case.\r<BR>\n" );
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document.write( "Let's say we placed point E at the origin (0,0).<BR>\n" );
document.write( "Let's place point A at (p,q) where p and q are any real numbers. I'll select p = 2 and q = 3 as shown in the diagram below.<BR>\n" );
document.write( "To go from E to A we go p units right and q units up.<BR>\n" );
document.write( "To go from E to C, we go in reverse: go p units left and q units down. This places C at (-p,-q)<BR>\n" );
document.write( "By construction, AE = EC to show that diagonal AC is bisected.<BR>\n" );
document.write( "You can use the distance formula, or Pythagorean theorem, to show that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AE+=+sqrt%28p%5E2%2Bq%5E2%29 ALIGN=MIDDLE  ALT=\"AE+=+sqrt%28p%5E2%2Bq%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=EC+=+sqrt%28p%5E2%2Bq%5E2%29 ALIGN=MIDDLE  ALT=\"EC+=+sqrt%28p%5E2%2Bq%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to arrive at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AE+=+EC ALIGN=MIDDLE  ALT=\"AE+=+EC\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's say point B is located at (r,s) where r and s are any real numbers. I'll pick r = 3 and s = 1.<BR>\n" );
document.write( "Follow similar logic as the previous paragraph to determine point D is located at (-r,-s). <BR>\n" );
document.write( "A similar construction leads to BE = ED to show that diagonal BD is bisected.\r<BR>\n" );
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document.write( "Key takeaway:<BR>\n" );
document.write( "A = (p,q)<BR>\n" );
document.write( "B = (r,s)<BR>\n" );
document.write( "C = (-p,-q)<BR>\n" );
document.write( "D = (-r,-s)<BR>\n" );
document.write( "E = (0,0)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Agraph%28400%2C400%2C-5%2C5%2C-5%2C5%29%2C%0D%0Acircle%282%2C3%2C0.08%29%2Ccircle%282%2C3%2C0.1%29%2Ccircle%282%2C3%2C0.12%29%2Ccircle%282%2C3%2C0.14%29%2Ccircle%283%2C1%2C0.08%29%2Ccircle%283%2C1%2C0.1%29%2Ccircle%283%2C1%2C0.12%29%2Ccircle%283%2C1%2C0.14%29%2Ccircle%28-2%2C-3%2C0.08%29%2Ccircle%28-2%2C-3%2C0.1%29%2Ccircle%28-2%2C-3%2C0.12%29%2Ccircle%28-2%2C-3%2C0.14%29%2Ccircle%28-3%2C-1%2C0.08%29%2Ccircle%28-3%2C-1%2C0.1%29%2Ccircle%28-3%2C-1%2C0.12%29%2Ccircle%28-3%2C-1%2C0.14%29%2Ccircle%280%2C0%2C0.08%29%2Ccircle%280%2C0%2C0.1%29%2Ccircle%280%2C0%2C0.12%29%2Ccircle%280%2C0%2C0.14%29%2C%0D%0Aline%282%2C3%2C3%2C1%29%2Cline%283%2C1%2C-2%2C-3%29%2Cline%28-2%2C-3%2C-3%2C-1%29%2Cline%28-3%2C-1%2C2%2C3%29%2Cline%282%2C3%2C-2%2C-3%29%2Cline%283%2C1%2C-3%2C-1%29%2C%0D%0Alocate%282.2%2C3.2%2C%22A%22%29%2Clocate%283.2%2C1.2%2C%22B%22%29%2Clocate%28-1.8%2C-2.8-0.2%2C%22C%22%29%2Clocate%28-2.8-0.5%2C-0.8-0.5%2C%22D%22%29%2Clocate%280.2%2C0.2-0.2%2C%22E%22%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In the diagram above we have p=2,q=3,r=3,s=1. <BR>\n" );
document.write( "However, you can select any four real numbers you want for p,q,r,s. \r<BR>\n" );
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document.write( "Compute the slope of side AB<BR>\n" );
document.write( "m = (y2-y1)/(x2-x1)<BR>\n" );
document.write( "m = (s-q)/(r-p)<BR>\n" );
document.write( "Compute the slope of side CD<BR>\n" );
document.write( "m = (y2-y1)/(x2-x1)<BR>\n" );
document.write( "m = (-s-(-q))/(-r-(-p))<BR>\n" );
document.write( "m = (-s+q)/(-r+p)<BR>\n" );
document.write( "m = (-1(s-q))/(-1(r-p))<BR>\n" );
document.write( "m = (s-q)/(r-p)<BR>\n" );
document.write( "Both slope results are the same expression. To avoid dividing by zero, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%3C%3Ep ALIGN=MIDDLE  ALT=\"r%3C%3Ep\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since slopeAB = slopeCD, we conclude that AB is parallel to CD.<BR>\n" );
document.write( "Similar steps are followed to prove that BC is parallel to AD.<BR>\n" );
document.write( "This concludes the proof that ABCD is a parallelogram.<BR>\n" );
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