document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=177791>Question 177791</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Analytic geometry <BR>\n" );
document.write( "Determine the shortest distance from the point H(5,2) to the line through points J(-6,4) and K(-2,-4). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #132851 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=132851\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find the shortest distance from point H to line JK you need to find a line perpendicular to JK that passes through point H. Find the intersection point between the two lines and then calculate the distance.<BR>\n" );
document.write( "First, find the slope of line JK. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BJK%5D=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29=%28-4-4%29%2F%28-2-%28-6%29%29=-8%2F4=-2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Perpendicular lines have slopes that are negative reciprocals of each other.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BJK%5Dm%5BP%5D=-1 ALIGN=MIDDLE  ALT=\"m%5BJK%5Dm%5BP%5D=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2m%5BP%5D=-1 ALIGN=MIDDLE  ALT=\"-2m%5BP%5D=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BP%5D=1%2F2 ALIGN=MIDDLE  ALT=\"m%5BP%5D=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Use the point slope form of the line using this slope and point H.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-y%5B0%5D=m%28x-x%5B0%5D%29 ALIGN=MIDDLE  ALT=\"y-y%5B0%5D=m%28x-x%5B0%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-2=%281%2F2%29%28x-5%29 ALIGN=MIDDLE  ALT=\"y-2=%281%2F2%29%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-2=%281%2F2%29x-5%2F2 ALIGN=MIDDLE  ALT=\"y-2=%281%2F2%29x-5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29x-1%2F2 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29x-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28+300%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C+grid%281%29%2C+circle%285%2C2%2C0.2%29%2Ccircle%28-6%2C4%2C0.2%29%2C%0D%0Agreen%28line%28-2%2C-4%2C-6%2C4%29%29%2C%0D%0Acircle%28-2%2C-4%2C-.2%29%2C+graph%28+300%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C+%281%2F2%29%28x-1%29%2C-2x-8%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next step is to find the intersection point of the two lines. <BR>\n" );
document.write( "The first line equation is,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-4=-2%28x-%28-6%29%29 ALIGN=MIDDLE  ALT=\"y-4=-2%28x-%28-6%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=y-4=-2x-12 ALIGN=MIDDLE  ALT=\"y-4=-2x-12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x-8 ALIGN=MIDDLE  ALT=\"y=-2x-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Set the two equations equal to each other and solve for x, then y.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29x-1%2F2=-2x-8 ALIGN=MIDDLE  ALT=\"%281%2F2%29x-1%2F2=-2x-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1=-4x-16 ALIGN=MIDDLE  ALT=\"x-1=-4x-16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x=-15 ALIGN=MIDDLE  ALT=\"5x=-15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x-8 ALIGN=MIDDLE  ALT=\"y=-2x-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2%28-3%29-8 ALIGN=MIDDLE  ALT=\"y=-2%28-3%29-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=6-8 ALIGN=MIDDLE  ALT=\"y=6-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2 ALIGN=MIDDLE  ALT=\"y=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The intersection between the two lines occurs at (-3,-2).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28+300%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C+grid%281%29%2C+circle%285%2C2%2C0.2%29%2Ccircle%28-6%2C4%2C0.2%29%2C%0D%0Agreen%28line%28-2%2C-4%2C-6%2C4%29%29%2C%0D%0Acircle%28-2%2C-4%2C-.2%29%2Ccircle%28-3%2C-2%2C.3%29%2C+graph%28+300%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C+%281%2F2%29%28x-1%29%2C-2x-8%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Finally, use the distance formula to calculate the distance from H to the intersection point.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2=%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2 ALIGN=MIDDLE  ALT=\"D%5E2=%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2=%28-3-5%29%5E2%2B%28-2-2%29%5E2 ALIGN=MIDDLE  ALT=\"D%5E2=%28-3-5%29%5E2%2B%28-2-2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2=%28-8%29%5E2%2B%28-4%29%5E2 ALIGN=MIDDLE  ALT=\"D%5E2=%28-8%29%5E2%2B%28-4%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2=64%2B16 ALIGN=MIDDLE  ALT=\"D%5E2=64%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2=80 ALIGN=MIDDLE  ALT=\"D%5E2=80\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=sqrt%2880%29 ALIGN=MIDDLE  ALT=\"D=sqrt%2880%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );