document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1154920>Question 1154920</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The diagonals of quadrilateral ABCD intersect at E(- 1, 4) . ABCD has vertices at A(1, 7) and B(- 4.5) What must be the coordinates of C and D to ensure 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 #777443 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=777443\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Start at point A(1,7). To move to point E(-1,4), we need to move 3 units down and 2 units to the left. We can see this in the diagram below.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"50%\" src= \"https://i.imgur.com/SvgaiBr.png\"><BR>\n" );
document.write( "Repeat this pattern of \"down 3, left 2\" one more time and you'll arrive at point C(-3,1)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"50%\" src= \"https://i.imgur.com/sil3wTn.png\">\r<BR>\n" );
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document.write( "Now focus on point B(-4, 5). To move from here to E(-1,4), we move 1 unit down and 3 units to the right. Do this again to arrive at point D (2,3)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"50%\" src= \"https://i.imgur.com/eJJraGx.png\">\r<BR>\n" );
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document.write( "These moves are valid because AE = EC, and BE = ED. Recall that the diagonals of any parallelogram always bisect each other. \"Bisect\" means to \"cut in half\". \r<BR>\n" );
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document.write( "You can go without using a visual diagram. Instead, you can use coordinate geometry. To go from A to E, we subtract 2 from the x coordinate and subtract 3 from the y coordinate. We can say the translation rule is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x,y) \to (x-2,y-3)\">. If you apply this translation rule to point E, then E moves to point C's location. Similarly, the translation rule of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x,y) \to (x+3,y-1)\"> is used to go from B to E. Use it on point E to move it to point D. \r<BR>\n" );
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document.write( "Final Diagram<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"50%\" src= \"https://i.imgur.com/QmawFLw.png\"><BR>\n" );
document.write( "To prove that we have a parallelogram, all you need to do is show that the opposite sides are parallel. To do this, compute the slopes of segments AB and CD. You should find the slopes to be equal. Parallel lines always have equal slopes. You should also find that segments BC and AD have equal slopes as well, so they are parallel also.\r<BR>\n" );
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document.write( "Answers:  <BR>\n" );
document.write( "<font color=red size=4>C is at (-3, 1)</font><BR>\n" );
document.write( "<font color=red size=4>D is at (2, 3)</font>\r<BR>\n" );
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