document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=291653>Question 291653</A>: <I><SPAN STYLE=\"word-break: break-all;\"> let A(-3,2) , B(-1,-3) and C(3,4) be the three vertices of a parallelogram, determine the coordinate of the fourth point.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #210876 by </B><A HREF=/tutors/aboutme.mpl?userid=CharlesG2><B>CharlesG2(834)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CharlesG2><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=210876\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let A(-3,2) , B(-1,-3) and C(3,4) be the three vertices of a parallelogram, determine the coordinate of the fourth point.<BR>\n" );
document.write( "A(-3,2)<BR>\n" );
document.write( "B(-1,-3)<BR>\n" );
document.write( "C(3,4)<BR>\n" );
document.write( "see what we got on a grid:<BR>\n" );
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document.write( "D(?,?), length DC = ?, length AD = ?<BR>\n" );
document.write( "AB and DC need to be equal length and parallel<BR>\n" );
document.write( "BC and AD need to be equal length and parallel<BR>\n" );
document.write( "parallel lines have the same slope<BR>\n" );
document.write( "AB = sqrt((-1 - -3)^2 + (-3 - 2)^2)<BR>\n" );
document.write( "AB = sqrt(2^2 + -5^2)<BR>\n" );
document.write( "AB = sqrt(4 + 25) = sqrt(29) needs to equal DC<BR>\n" );
document.write( "BC = sqrt((3 - -1)^2 + (4 - -3)^2)<BR>\n" );
document.write( "BC = sqrt(4^2 + 7^2)<BR>\n" );
document.write( "BC = sqrt(16 + 49) = sqrt(65) needs to equal AD<BR>\n" );
document.write( "slope AB = (-3 - 2)/(-1 - -3) = -5/2 needs to be same as slope DC<BR>\n" );
document.write( "slope BC = (4 - -3)/(3 - -1) = 7/4 needs to be same as slope AD<BR>\n" );
document.write( "call x and y the x and y coordinates of D<BR>\n" );
document.write( "AB = DC = sqrt(29) = sqrt((x - 3)^2 + (y - 4)^2)<BR>\n" );
document.write( "29 = x^2 - 6x + 9 + y^2 - 8y + 16<BR>\n" );
document.write( "slope AB = slope DC = -5/2 = (y - 4)/(x - 3)<BR>\n" );
document.write( "-5(x - 3) = 2(y - 4)<BR>\n" );
document.write( "-5x + 15 = 2y - 8<BR>\n" );
document.write( "-5x - 2y = -23<BR>\n" );
document.write( "-5x + 23 = 2y<BR>\n" );
document.write( "(-5/2)x + (23/2) = y<BR>\n" );
document.write( "BC = AD = sqrt(65) = sqrt((x + 3)^2 + (y - 2)^2)<BR>\n" );
document.write( "65 = x^2 + 6x + 9 + y^2 - 4y + 4<BR>\n" );
document.write( "slope BC = slope AD = 7/4 = (y - 2)/(x + 3)<BR>\n" );
document.write( "7(x + 3) = 4(y - 2)<BR>\n" );
document.write( "7x + 21 = 4y - 8<BR>\n" );
document.write( "7x - 4y = -29<BR>\n" );
document.write( "7x + 29 = 4y<BR>\n" );
document.write( "(7/4)x + (29/4) = y<BR>\n" );
document.write( "system of equations:<BR>\n" );
document.write( "-5x - 2y = -23 (multiply by -2 and add to 2nd) --> 10x + 4y = 46<BR>\n" );
document.write( " 7x - 4y = -29<BR>\n" );
document.write( "-->  -5x - 2y = -23<BR>\n" );
document.write( "     17x + 0y = 17<BR>\n" );
document.write( "x = 1 --> -5 - 2y = -23 --> -2y = -18 --> y = 9<BR>\n" );
document.write( "so D is (1,9)<BR>\n" );
document.write( "checking: slope AB = slope DC = -5/2 = (y - 4)/(x - 3) = 5/(-2) = -5/2<BR>\n" );
document.write( "          slope BC = slope AD = 7/4 = (y - 2)/(x + 3) = 7/4<BR>\n" );
document.write( "          AB = DC = sqrt(29) = sqrt((x - 3)^2 + (y - 4)^2) =<BR>\n" );
document.write( "                             = sqrt(-2^2 + 5^2)<BR>\n" );
document.write( "                             = sqrt(4 + 25) = sqrt(29)<BR>\n" );
document.write( "          BC = AD = sqrt(65) = sqrt((x + 3)^2 + (y - 2)^2) =<BR>\n" );
document.write( "                             = sqrt(4^2 + 7^2)<BR>\n" );
document.write( "                             = sqrt(16 + 49) = sqrt(65)<BR>\n" );
document.write( "so D is indeed (1,9)\r<BR>\n" );
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