document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=186973>Question 186973</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For A(-2,-4), B(-5,-1), and C(0,-4). find all locations of a fourth point, D, so that a paralleogram is formed using A, B, C, D in any order as vertices. Plot each point D on a coodinate grid and draw the parallelogram. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #140167 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=140167\">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( "Since you are given 3 points, you can select 3 pairs of points.  Using the two-point form of the equation of a line, you can create three equations representing three lines.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y - y_1 = \left(\frac{y_1 - y_2}{x_1 - x_2}\right)(x - x_1) \">\r<BR>\n" );
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document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_1(-2,-4)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_2(-5,-1)\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ L_1: y + 4 = \left(\frac{-4 + 1}{-2 + 5}\right)(x + 2) \ \ \Rightarrow\ \ y = -x -6\"> \r<BR>\n" );
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document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_2(-5,-1)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_3(0,-4)\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ L_2: y + 1 = \left(\frac{-1 + 4}{-5 - 0}\right)(x + 5) \ \ \Rightarrow\ \ y = -\frac{3}{5}x -4\">\r<BR>\n" );
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document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_1(-2,-4)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_3(0,-4)\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ L_3: y + 4 = \left(\frac{-4 + 4}{-2 - 0}\right)(x + 2) \ \ \Rightarrow\ \ y = - 4\">\r<BR>\n" );
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document.write( "Now, using:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  L_\alpha \parallel L_\beta \ \ \Leftrightarrow\ \ m_\alpha = m_\beta\">\r<BR>\n" );
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document.write( "Create the equation of the parallel to each of the above lines through the given point <i><b>not</b></i> on that line using the point-slope form of the equation of a line:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large m_1 = -1\"> through <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_3(0,-4):\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ L_4: y + 4 = -1(x - 0) \ \ \Rightarrow\ \ y = -x -4\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large m_2 = -\frac{3}{5}\"> through <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_2(-5,-1):\">\r<BR>\n" );
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document.write( "(Left as an exercise for the student)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large m_3 = 0\"> through <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_2(-5,-1):\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ L_6: y + 1 = 0(x + 5) \ \ \Rightarrow\ \ y = -1\">\r<BR>\n" );
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document.write( "Now that we have <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L_4 \parallel L_1\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L_6 \parallel L_3\"> we only need to find the point of intersection between <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L_4\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L_6\">\r<BR>\n" );
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document.write( "Since we know that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y=-1\">\r<BR>\n" );
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document.write( "from the equation for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L_6\">, we can just substitute into the equation for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L_4\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \   -1 = -x -4 \ \ \Rightarrow\ \ x = -3\">\r<BR>\n" );
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document.write( "Hence, Point D for the first parallelogram is (-3,-1).\r<BR>\n" );
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document.write( "Now all you have to do is derive the equation for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L_5\"> and find the points of intersection with the other two lines to get the other two coordinate pairs for the other two possibilities for Point D.\r<BR>\n" );
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document.write( "John<BR>\n" );
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