document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=669172>Question 669172</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the slope of two diagonals of quadrilateral whose vertices are (3,1), (-1,2), (-3,-1) and (1,-2). identify the quadrilateral. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #416166 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=416166\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the first diagonal goes from (-1,2) to (1,-2)<BR>\n" );
document.write( "the slope of this diagonal would be change in y divided by change in x which becomes (-2-2) / (1-(-1)) which becomes -4 / 2 which becomes -2.\r<BR>\n" );
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document.write( "the second diagonal goes from (-3,-1) to (3,1)<BR>\n" );
document.write( "the slope of this diagonal would be change in y divided by change in x which becomes (1 - (-1)) / (3 - (-3)) which becomes 2 / 6 which becomes 1/3.\r<BR>\n" );
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document.write( "it looks like we have a parallelogram.<BR>\n" );
document.write( "the diagonals appear to bisect each other.<BR>\n" );
document.write( "the intersection point appears to be (0,0).\r<BR>\n" );
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document.write( "the point (1,-2) is a reflection of the point (-1,2) about the origin because (x,y) becomes (-x,-y).<BR>\n" );
document.write( "the point (3,1) is a reflection of the point (-3,-1) about the origin because (x,y) becomes (-x,-y).\r<BR>\n" );
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document.write( "being reflections about the origin means that the the distance from the origin of both pairs of points is the same for each pair.\r<BR>\n" );
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document.write( "this means the length of (0,0) to (1,-2) is the same length as (0,0) to (-1,2).<BR>\n" );
document.write( "this also means the length of (0,0) to (3,1) is  the same length as (0,0) to (-3,-1).\r<BR>\n" );
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document.write( "since a quadrilateral is a parallelogram if the diagonals bisect each other, this should be sufficient proof.\r<BR>\n" );
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document.write( "bottom line is you found the slope as requested.<BR>\n" );
document.write( "you determined from the points themselves that the ends of each diagonal are reflections about the origin.<BR>\n" );
document.write( "you deduced from this that the diagonals crossed at the origin (0,0).<BR>\n" );
document.write( "you deduced from this that the distance from the points to the origin had to be the same for each diagonal point pair.<BR>\n" );
document.write( "since the diagonals bisect each other, you deduced that the quadrilateral had to be a parallelogram based on a geometric theorem previously proven that states that.\r<BR>\n" );
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