document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=116051>Question 116051</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Floor plans for a building have the four corners of a room located at the points (2,3),(11,6), (-3,18), and (8,21). Determine whether the side through the points (2,3), and (11,6) is parallel to the side through the points (-3,18) and (8,21).<BR>\n" );
document.write( "Determine whether the side through the points (2,3) and (11,6) is perpendicular to the side through the points (2,3) and (-3,18).<BR>\n" );
document.write( "Help,please.\r<BR>\n" );
document.write( "\n" );
document.write( "Thank you, barb neely </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #84403 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=84403\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hello again, Barb,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The principles that are being illustrated in this problem are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1) Creating equations of lines using the 2-point form of the line.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2) Converting these equations to the point-slope form of the line.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3) Comparing the slopes of lines to determine parallelism and perpendicularity.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First, you need equations of the three lines that we'll call lines A, B, and C.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Using the two-point form of the line:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-y%5B1%5D%29=%28%28y%5B1%5D-y%5B2%5D%29%2F%28x%5B1%5D-x%5B2%5D%29%29%28x-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"%28y-y%5B1%5D%29=%28%28y%5B1%5D-y%5B2%5D%29%2F%28x%5B1%5D-x%5B2%5D%29%29%28x-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Line A through points (2,3) and (11, 6).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-3%29=%28%283-6%29%2F%282-11%29%29%28x-2%29 ALIGN=MIDDLE  ALT=\"%28y-3%29=%28%283-6%29%2F%282-11%29%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and do the arithmetic to get the point-slope form:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-3%29=%28-3%2F-9%29%28x-2%29 ALIGN=MIDDLE  ALT=\"%28y-3%29=%28-3%2F-9%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-3%29=%281%2F3%29%28x-2%29 ALIGN=MIDDLE  ALT=\"%28y-3%29=%281%2F3%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BA%5D=1%2F3 ALIGN=MIDDLE  ALT=\"m%5BA%5D=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now, Line B from (-3,18) to (8,21)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-18%29=%28%2818-21%29%2F%28-3-8%29%29%28x-%28-3%29%29 ALIGN=MIDDLE  ALT=\"%28y-18%29=%28%2818-21%29%2F%28-3-8%29%29%28x-%28-3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-18%29=%28%28-3%29%2F%28-11%29%29%28x%2B3%29 ALIGN=MIDDLE  ALT=\"%28y-18%29=%28%28-3%29%2F%28-11%29%29%28x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BB%5D=3%2F11 ALIGN=MIDDLE  ALT=\"m%5BB%5D=3%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Finally, Line C from (2,3) to (-3,18)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-3%29=%28%283-18%29%2F%282-%28-3%29%29%29%28x-2%29 ALIGN=MIDDLE  ALT=\"%28y-3%29=%28%283-18%29%2F%282-%28-3%29%29%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-3%29=%28%28-15%29%2F%285%29%29%28x-2%29 ALIGN=MIDDLE  ALT=\"%28y-3%29=%28%28-15%29%2F%285%29%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-3%29=%28%28-3%29%2F1%29%28x-2%29 ALIGN=MIDDLE  ALT=\"%28y-3%29=%28%28-3%29%2F1%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BC%5D=-3 ALIGN=MIDDLE  ALT=\"m%5BC%5D=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now, your two questions are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Is line A parallel to line B?  Lines are parallel if and only if their slopes are equal.  So is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BA%5D=m%5BB%5D ALIGN=MIDDLE  ALT=\"m%5BA%5D=m%5BB%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >?  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3%3C%3E3%2F11 ALIGN=MIDDLE  ALT=\"1%2F3%3C%3E3%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" >, therefore the lines are NOT parallel.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Is line A perpendicular to line C?  Lines are perpendicular if and only if their slopes are negative reciprocals of each other.  So does <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BA%5D=%28-1%29%2Fm%5BB%5D ALIGN=MIDDLE  ALT=\"m%5BA%5D=%28-1%29%2Fm%5BB%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >?  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3=%28-1%29%2F%28-3%29 ALIGN=MIDDLE  ALT=\"1%2F3=%28-1%29%2F%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, therefore the lines ARE perpendicular.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Drawing a diagram may show you that this result makes sense:<BR>\n" );
document.write( "Line A is the red one, Line B is the green one, Line C is the blue one.  You can see that A and B are almost, but not quite parallel, and A and C look perpendicular which are the results that we got algebraically.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-8%2C15%2C-2%2C22%2Cx%2F3%2B7%2F3%2C3x%2F11%2B207%2F11%2C-3x%2B9%2C-3x%2B39%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Hope that helps,<BR>\n" );
document.write( "John </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );