document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=978962>Question 978962</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Math Question:\r<BR>\n" );
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document.write( "Prove using coordinate geometry:<BR>\n" );
document.write( "If you connect successive midpoints of a rectangle, the resulting figure is a rhombus.\r<BR>\n" );
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document.write( "(Our teacher isn't the type to explain things at all in class so I have no idea what/how to solve this) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #600346 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=600346\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "I call bullshit on the \"teacher isn't the type...\" nonsense.  Your teacher gets paid a salary, said funds either derived from the tuition you pay to go to school or the property taxes your parents pay to run the public schools.  Your teacher therefore has a fiduciary responsibility to \"explain things\".  So cut the whiny \"poor me\" crap and use your energy to learn mathematics.\r<BR>\n" );
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document.write( "Be that as it may, on a coordinate system, create an arbitrary rectangle of dimensions <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\">.  Since coordinates can be translated and rotated arbitrarily, you can position the rectangle with one vertex at the origin and the sides parallel to the coordinate axes without loss of generality.  Hence, your vertices are (given <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> is the horizontal dimension), <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (0,0), (a,0), (a,b), (0,b)\">.  Using the midpoint formulas:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_m\ = \frac{x_1 + x_2}{2}\"> and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y_m\ = \frac{y_1 + y_2}{2}\">\r<BR>\n" );
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document.write( "calculate the coordinates of each of the midpoints of the four sides of the rectangle, in terms of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\">.  For example, the side between the points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (a,0)\"> and  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (a,b)\"> has a midpoint <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (a,\frac{b}{2})\">.\r<BR>\n" );
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document.write( "Then, using the distance formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ d\ =\ \sqrt{(x_1\ -\ x_2)^2\ +\ (y_1\ -\ y_2)^2}\">\r<BR>\n" );
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document.write( "calculate the distance between each pair of adjacent side midpoints.  If the inscribed figure is a rhombus, all of the sides must be equal.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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