document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=998307>Question 998307</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the points A(-7, 10), B(12, 7), C(10, -24), and D(-8, -3). Which two lines determined by these points are perpendicular? Explain. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #616122 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=616122\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If two lines are perpendicular the product of their slopes is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With those 4 points, we can make<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A3=12 ALIGN=MIDDLE  ALT=\"4%2A3=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > rays, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F2=6 ALIGN=MIDDLE  ALT=\"12%2F2=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > lines.<BR>\n" );
document.write( "We can calculate the slope of all those lines,<BR>\n" );
document.write( "but that is a lot of work.<BR>\n" );
document.write( "With some graph paper, to save time and effort, we could plot the 4 points and graph the 6 lines.<BR>\n" );
document.write( "That would let us know which ones may be perpendicular,<BR>\n" );
document.write( "and which definitely are not perpendicular.<BR>\n" );
document.write( "Then, we can calculate the slopes of online the lines that might be perpendicular, and find which ones really are perpendicular.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C400%2C-12%2C18%2C-27%2C13%2Cgrid%281%29%2C%0D%0Agreen%28line%28-7%2C10%2C12%2C7%29%29%2Cgreen%28line%28-7%2C10%2C10%2C-24%29%29%2C%0D%0Agreen%28line%28-7%2C10%2C-8%2C-3%29%29%2Cgreen%28line%2812%2C7%2C-8%2C-3%29%29%2C%0D%0Agreen%28line%2812%2C7%2C10%2C-24%29%29%2Cgreen%28line%28-8%2C-3%2C10%2C-24%29%29%2C%0D%0Ared%28circle%28-7%2C10%2C0.4%29%29%2Clocate%28-10.5%2C12%2Cred%28A%28-7%2C10%29%29%29%2C%0D%0Ared%28circle%2812%2C7%2C0.4%29%29%2Clocate%2812.5%2C9%2Cred%28B%2812%2C7%29%29%29%2C%0D%0Ared%28circle%2810%2C-24%2C0.4%29%29%2Clocate%289.5%2C-24.5%2Cred%28C%2810%2C-24%29%29%29%2C%0D%0Ared%28circle%28-8%2C-3%2C0.4%29%29%2Clocate%28-10.5%2C-4%2Cred%28D%28-8%2C-3%29%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Lines AB and BC seem almost perpendicular, but AC and BD look perpendicular.<BR>\n" );
document.write( "Let's check the slopes of AC and BD.<BR>\n" );
document.write( "The slope of AC is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BAC%5D=%28y%5BA%5D-y%5BC%5D%29%2F%28x%5BA%5D-x%5BC%5D%29=%2810-%28-24%29%29%2F%28-7-10%29=%2810%2B24%29%2F%28-17%29=34%2F%28-17%29=-2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The slope of BD is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BBD%5D=%28y%5BB%5D-y%5BD%5D%29%2F%28x%5BB%5D-x%5BD%5D%29=%287-%28-3%29%29%2F%2812-%28-8%29%29=%287%2B3%29%2F%2812%2B8%29=10%2F20=1%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The product of the slopes is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%29%2A%281%2F2%29=-1 ALIGN=MIDDLE  ALT=\"%28-2%29%2A%281%2F2%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so AC and BD are indeed perpendicular. </SPAN>\n" );
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