document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=187159>Question 187159</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Apply Slope, Midpoint and Length Formulas<BR>\n" );
document.write( "5. The vertices of a quadrilateral are K(-1, 0), L(1, -2), M(4, 1), and N(2, 3). Verify that<BR>\n" );
document.write( "a) KLMN is a rectangle <BR>\n" );
document.write( "b) the lengths of the diagonals of KLMN are equal<BR>\n" );
document.write( "Can you please help me with this question? thank you!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #140302 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=140302\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll give you hints to solve the problem. Let me know if you need further help\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To prove that any quadrilateral is a rectangle, you need to show that ALL of its angles are 90 degrees. Or stated another way, you need to show that the segments are perpendicular to one another.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To find out if two segments are perpendicular to one another, you need to find the slopes of the line segments. So you need to find the slope of the segments KL, LM, MN, and NK\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: the slope formula is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "From there, if you find that the product of the slopes KL and LM are -1, then this will show that KL is perpendicular to LM. If you do this to every paired segment, and you find that they're perpendicular, then this will prove that KLMN is a rectangle\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "b)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If you draw out rectangle KLMN, then you'll see that the diagonals are KM and NL\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now simply use the distance formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to  find the distance from points K and M. This distance should be equal to the distance from L to N. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );