document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=821011>Question 821011</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Three of the vertices of a parallelogram are (2, 1) (4, 7), and (6, 5).<BR>\n" );
document.write( "What are the three possible coordinates of the fourth vertex? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #493887 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=493887\">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%28300%2C300%2C-4%2C16%2C-4%2C16%2C%0D%0Agrid%281%29%2Cred%28circle%282%2C1%2C0.4%29%29%2C%0D%0Ared%28circle%284%2C7%2C0.4%29%29%2Cred%28circle%286%2C5%2C0.4%29%29%2C%0D%0Agreen%28circle%280%2C3%2C0.4%29%29%2C+blue%28circle%288%2C11%2C0.4%29%29%2C%0D%0Ared%28triangle%282%2C1%2C4%2C7%2C6%2C5%29%29%2C%0D%0Ablue%28line%288%2C11%2C4%2C7%29%29%2Cblue%28line%288%2C11%2C6%2C5%29%29%2C%0D%0Agreen%28line%280%2C3%2C2%2C1%29%29%2Cgreen%28line%280%2C3%2C4%2C7%29%29%2C%0D%0Acircle%284%2C-1%2C0.4%29%2Cline%284%2C-1%2C2%2C1%29%2Cline%284%2C-1%2C6%2C5%29%2C%0D%0Alocate%28-1%2C4%2CX%29%2Clocate%288.4%2C12%2CY%29%2Clocate%284.5%2C-1%2CZ%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The three given points are in red,and I connected them into a red triangle.<BR>\n" );
document.write( "Parallelogram sides parallel to those 3 red sides,form a similar triangle and the vertices of that larger triangle are the possible fourth vertices for the parallelogram.<BR>\n" );
document.write( "Point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%280%2C3%29 ALIGN=MIDDLE  ALT=\"X%280%2C3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > completes the green (and red) parallelogram, which happens to be a rectangle.<BR>\n" );
document.write( "Point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y%288%2C11%29 ALIGN=MIDDLE  ALT=\"Y%288%2C11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > completes the blue (and red) parallelogram.<BR>\n" );
document.write( "Point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Z%284%2C-1%29 ALIGN=MIDDLE  ALT=\"Z%284%2C-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > completes the black (and red) parallelogram.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Drawing is easier than calculating coordinates.<BR>\n" );
document.write( "In the parallelogram, one of the 3 given points must be connected to both of the other ones, and the other ones form the diagonal of the parallelogram.<BR>\n" );
document.write( "If point (6,5) is connected to (2, 1) and (4, 7),<BR>\n" );
document.write( "the line connecting (2, 1) and (4, 7) is a diagonal of the parallelogram.<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28A%29 ALIGN=MIDDLE  ALT=\"red%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28B%29 ALIGN=MIDDLE  ALT=\"blue%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28C%29 ALIGN=MIDDLE  ALT=\"red%28C%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28D%29 ALIGN=MIDDLE  ALT=\"blue%28D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a parallelogram, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28AC%29 ALIGN=MIDDLE  ALT=\"red%28AC%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a diagonal.<BR>\n" );
document.write( "To go from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28A%29 ALIGN=MIDDLE  ALT=\"red%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the fourth vertex, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28D%29 ALIGN=MIDDLE  ALT=\"blue%28D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we go on the parallel side opposite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28B%29 ALIGN=MIDDLE  ALT=\"blue%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28C%29 ALIGN=MIDDLE  ALT=\"red%28C%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the same distance and direction as from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28B%29 ALIGN=MIDDLE  ALT=\"blue%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28C%29 ALIGN=MIDDLE  ALT=\"red%28C%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Going from A(4,7) to one 4th vertex, is the same as from B(6,5) to C(2,1),<BR>\n" );
document.write( "4 units left and 4 down, and we find X(0,3).<BR>\n" );
document.write( "Going from A(4,7) to the 4th vertex, is the same as from B(2,1) to C(6,5),<BR>\n" );
document.write( "4 units right and 4 up, and we find Y(8,11).<BR>\n" );
document.write( "Going from A(2,1) to the 4th vertex, is the same as from B(4,7) to C(6,5),<BR>\n" );
document.write( "2 units right and 2 down, and we find Z(4,-1). </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );