document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=283748>Question 283748</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A square has vertices U(-2,1), V(2,3), W(4,-1) and X(0,-3). Verify that the diagonals perpendicularly bisect each other. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #205929 by </B><A HREF=/tutors/aboutme.mpl?userid=oberobic><B>oberobic(2304)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=oberobic><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=205929\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The four points can be plotted:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0A++grid%281%29%2C%0D%0A++blue%28circle%28-2%2C1%2C.15%29%29%2C%0D%0A++blue%28circle%282%2C3%2C.15%29%29%2C%0D%0A++blue%28circle%284%2C-1%2C.15%29%29%2C%0D%0A++blue%28circle%280%2C-3%2C.15%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The diagonals are defined by the points: U,W and V,X.<BR>\n" );
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document.write( "We need to solve for a line going through the first pair of points<BR>\n" );
document.write( "U (-2,1)<BR>\n" );
document.write( "W (4,-1)<BR>\n" );
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document.write( "y = mx + b<BR>\n" );
document.write( "m= (-1-1)/(4-(-2))= -2/6 = -1/3<BR>\n" );
document.write( "y = -1/3x + b<BR>\n" );
document.write( "We can solve for 'b' by plugging in the points we know.<BR>\n" );
document.write( "1 = -1/3*(-2 ) + b<BR>\n" );
document.write( "b = 1/3<BR>\n" );
document.write( "or<BR>\n" );
document.write( "-1 = -1/3*4 +b<BR>\n" );
document.write( "b = 1/3<BR>\n" );
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document.write( "y=-1/3x + 1/3  defines the equation for the first diagonal.<BR>\n" );
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document.write( "Then we need to find the equation for the line going through the other two points<BR>\n" );
document.write( "V (2,3)<BR>\n" );
document.write( "X (0,-3)<BR>\n" );
document.write( "m= (-3-3)/(0-2) = -6/-2 = 6/2 = 3<BR>\n" );
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document.write( "At this point we know the diagonals are perpendicular because, by definition, the slopes of perpendicular lines are the negative reciprocal.<BR>\n" );
document.write( "-1/3 is the negative reciprocal of 3.<BR>\n" );
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document.write( "But we can press on to continue to define the second equation...<BR>\n" );
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document.write( "-3 = 3(0) + b<BR>\n" );
document.write( "b = -3<BR>\n" );
document.write( "or<BR>\n" );
document.write( "3 = 3(2) + b<BR>\n" );
document.write( "b = -3<BR>\n" );
document.write( "y = 3x -3<BR>\n" );
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document.write( "Then a graph will illustrate they are perpendicular.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0A++grid%281%29%2C%0D%0A++graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-1%2F3%2Ax+%2B+1%2F3%2C3%2Ax-3%29%2C%0D%0A++blue%28circle%28-2%2C1%2C.15%29%29%2C%0D%0A++blue%28circle%282%2C3%2C.15%29%29%2C%0D%0A++blue%28circle%284%2C-1%2C.15%29%29%2C%0D%0A++blue%28circle%280%2C-3%2C.15%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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