document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=127642>Question 127642</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2. Prove that the points (-2, 9), (-4, -2), (1, -12) and (3, -1) are vertices of a rhombus (a parallelogram with all sides equal length). Show that the diagonals are perpendicular. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #93524 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=93524\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-15%2C15%2C-15%2C15%2C%0D%0Agraph%28500%2C500%2C-15%2C15%2C-15%2C15%2C0%29%2C%0D%0Acircle%28-2%2C9%2C0.08%29%2C%0D%0Acircle%28-2%2C9%2C0.1%29%2C%0D%0Acircle%28-2%2C9%2C0.12%29%2C%0D%0Acircle%28-4%2C-2%2C0.08%29%2C%0D%0Acircle%28-4%2C-2%2C0.1%29%2C%0D%0Acircle%28-4%2C-2%2C0.12%29%2C%0D%0Acircle%281%2C-12%2C0.08%29%2C%0D%0Acircle%281%2C-12%2C0.1%29%2C%0D%0Acircle%281%2C-12%2C0.12%29%2C%0D%0Acircle%283%2C-1%2C0.08%29%2C%0D%0Acircle%283%2C-1%2C0.1%29%2C%0D%0Acircle%283%2C-1%2C0.12%29%2C%0D%0Aline%28-2%2C9%2C-4%2C-2%29%2C%0D%0Aline%28-4%2C-2%2C1%2C-12%29%2C%0D%0Aline%281%2C-12%2C3%2C-1%29%2C%0D%0Aline%283%2C-1%2C-2%2C9%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "First let's find the slope of the line through the points (-2,9) and (1,-12) (these vertices are opposite from one other)\r<BR>\n" );
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document.write( "Let's denote the first point (-2,9) as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_{1},y_{1}\right)\">. In other words, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x_{1}=-2\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y_{1}=9\">\r<BR>\n" );
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document.write( "Now let's denote the second point (1,-12) as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_{2},y_{2}\right)\">. In other words, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x_{2}=1\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y_{2}=-12\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the slope formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%28-12-9%29%2F%281--2%29 ALIGN=MIDDLE  ALT=\"m=%28-12-9%29%2F%281--2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D=-12 ALIGN=MIDDLE  ALT=\"y%5B2%5D=-12\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=9 ALIGN=MIDDLE  ALT=\"y%5B1%5D=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=1 ALIGN=MIDDLE  ALT=\"x%5B2%5D=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=-2 ALIGN=MIDDLE  ALT=\"x%5B1%5D=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-21%2F3 ALIGN=MIDDLE  ALT=\"m=-21%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract the terms in the numerator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12-9 ALIGN=MIDDLE  ALT=\"-12-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-21 ALIGN=MIDDLE  ALT=\"-21\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Subtract the terms in the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1--2 ALIGN=MIDDLE  ALT=\"1--2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-7 ALIGN=MIDDLE  ALT=\"m=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "So the slope of the line through the points (-2,9) and (1,-12) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-7 ALIGN=MIDDLE  ALT=\"m=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's find the slope of the line through the points (-4,-2) and (3,-1) (these vertices are opposite from one other)\r<BR>\n" );
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document.write( "Let's denote the first point (-4,-2) as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_{1},y_{1}\right)\">. In other words, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x_{1}=-4\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y_{1}=-2\">\r<BR>\n" );
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document.write( "Now let's denote the second point (3,-1) as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_{2},y_{2}\right)\">. In other words, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x_{2}=3\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y_{2}=-1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the slope formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%28-1--2%29%2F%283--4%29 ALIGN=MIDDLE  ALT=\"m=%28-1--2%29%2F%283--4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D=-1 ALIGN=MIDDLE  ALT=\"y%5B2%5D=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=-2 ALIGN=MIDDLE  ALT=\"y%5B1%5D=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=3 ALIGN=MIDDLE  ALT=\"x%5B2%5D=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=-4 ALIGN=MIDDLE  ALT=\"x%5B1%5D=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=1%2F7 ALIGN=MIDDLE  ALT=\"m=1%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract the terms in the numerator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1--2 ALIGN=MIDDLE  ALT=\"-1--2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Subtract the terms in the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3--4 ALIGN=MIDDLE  ALT=\"3--4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the slope of the line through the points (-4,-2) and (3,-1) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=1%2F7 ALIGN=MIDDLE  ALT=\"m=1%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since the product of the two slopes is -1 (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7%281%2F7%29=-7%2F7=-1 ALIGN=MIDDLE  ALT=\"-7%281%2F7%29=-7%2F7=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >), this shows us that the two slopes are perpendicular. So the two diagonals are also perpendicular. This shows us that the figure is a rhombus. </SPAN>\n" );
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