document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1137650>Question 1137650</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the coordinates of the orthocentre of triangle DEF with vertices at D (-3,4) E (8,-2) F (4,5) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #755583 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=755583\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "given: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DEF ALIGN=MIDDLE  ALT=\"DEF\"   DISABLED_style= \"max-width:90%; height:auto;\" > with vertices at <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >) <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E ALIGN=MIDDLE  ALT=\"E\"   DISABLED_style= \"max-width:90%; height:auto;\" > (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F ALIGN=MIDDLE  ALT=\"F\"   DISABLED_style= \"max-width:90%; height:auto;\" > (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" >) \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the orthocenter: the point where the three \"altitudes\" of a triangle meet\r<BR>\n" );
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document.write( "find  the slopes of  the lines that contain segments <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DE ALIGN=MIDDLE  ALT=\"DE\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+EF ALIGN=MIDDLE  ALT=\"+EF\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DE+=+%28-2-4%29%29%2F%288-%28-3%29%29+=+-6%2F11+ ALIGN=MIDDLE  ALT=\"DE+=+%28-2-4%29%29%2F%288-%28-3%29%29+=+-6%2F11+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "now, we need a slope of perpendicular line, altitude to segments <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DE ALIGN=MIDDLE  ALT=\"DE\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "slope of perpendicular line, altitude :  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Fm+=+-1%2F%28-6%2F11%29 ALIGN=MIDDLE  ALT=\"-1%2Fm+=+-1%2F%28-6%2F11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > -> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+11%2F6 ALIGN=MIDDLE  ALT=\"m+=+11%2F6\"   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+=+mx+%2B+b ALIGN=MIDDLE  ALT=\"y+=+mx+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > (substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+11%2F6 ALIGN=MIDDLE  ALT=\"m+=+11%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4 ALIGN=MIDDLE  ALT=\"x+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+5 ALIGN=MIDDLE  ALT=\"y+=+5\"   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=5+=+%2811%2F6%29%284%29+%2B+b ALIGN=MIDDLE  ALT=\"5+=+%2811%2F6%29%284%29+%2B+b\"   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=5+=+44%2F6+%2B+b ALIGN=MIDDLE  ALT=\"5+=+44%2F6+%2B+b\"   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=b+=+5-22%2F3 ALIGN=MIDDLE  ALT=\"b+=+5-22%2F3\"   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=b+=+15%2F3-22%2F3 ALIGN=MIDDLE  ALT=\"b+=+15%2F3-22%2F3\"   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=b=+-7%2F3 ALIGN=MIDDLE  ALT=\"b=+-7%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "equation of the altitude to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DE ALIGN=MIDDLE  ALT=\"DE\"   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+=++%2811%2F6%29x+-7%2F3 ALIGN=MIDDLE  ALT=\"+y+=++%2811%2F6%29x+-7%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=EF+=+%28-2-4%29%2F%288-5%29+=+-7%2F4+ ALIGN=MIDDLE  ALT=\"EF+=+%28-2-4%29%2F%288-5%29+=+-7%2F4+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "altitude to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=EF ALIGN=MIDDLE  ALT=\"EF\"   DISABLED_style= \"max-width:90%; height:auto;\" > : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-1%2Fm+=+-1%2F%28-7%2F4%29 ALIGN=MIDDLE  ALT=\"+-1%2Fm+=+-1%2F%28-7%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > -> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+4%2F7 ALIGN=MIDDLE  ALT=\"m+=+4%2F7\"   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+=+mx+%2B+b ALIGN=MIDDLE  ALT=\"y+=+mx+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > (substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+4%2F7 ALIGN=MIDDLE  ALT=\"m+=+4%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <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;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=4 ALIGN=MIDDLE  ALT=\"y+=4\"   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=4=+-3%284%2F7+%29+%2B+b ALIGN=MIDDLE  ALT=\"4=+-3%284%2F7+%29+%2B+b\"   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=4+=+-12%2F7+%2B+b ALIGN=MIDDLE  ALT=\"4+=+-12%2F7+%2B+b\"   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=b+=+4%2B12%2F7 ALIGN=MIDDLE  ALT=\"b+=+4%2B12%2F7\"   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=b=+40%2F7 ALIGN=MIDDLE  ALT=\"b=+40%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "equation of the altitude to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=EF ALIGN=MIDDLE  ALT=\"EF\"   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+=+%284%2F7%29+x+%2B40%2F7 ALIGN=MIDDLE  ALT=\"y+=+%284%2F7%29+x+%2B40%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " \r<BR>\n" );
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document.write( "intersection of the altitudes will be orthocenter\r<BR>\n" );
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document.write( "equation 1:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=++%2811%2F6%29x+-7%2F3 ALIGN=MIDDLE  ALT=\"y+=++%2811%2F6%29x+-7%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "equation 2:   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=%284%2F7%29+x+%2B40%2F7 ALIGN=MIDDLE  ALT=\"y+=%284%2F7%29+x+%2B40%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Solving for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   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=%2811%2F6%29x+-7%2F3=%284%2F7%29+x+%2B40%2F7 ALIGN=MIDDLE  ALT=\"%2811%2F6%29x+-7%2F3=%284%2F7%29+x+%2B40%2F7\"   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=%2811%2F6%29x+-%284%2F7%29+x+=+7%2F3%2B40%2F7 ALIGN=MIDDLE  ALT=\"%2811%2F6%29x+-%284%2F7%29+x+=+7%2F3%2B40%2F7\"   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=%2853+x%29%2F42=+169%2F21 ALIGN=MIDDLE  ALT=\"%2853+x%29%2F42=+169%2F21\"   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=x+=+338%2F53 ALIGN=MIDDLE  ALT=\"x+=+338%2F53\"   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=x=+6.4 ALIGN=MIDDLE  ALT=\"x=+6.4\"   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+=++%2811%2F6%29x+-7%2F3 ALIGN=MIDDLE  ALT=\"y+=++%2811%2F6%29x+-7%2F3\"   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+=++%2811%2F6%29%286.4%29+-7%2F3 ALIGN=MIDDLE  ALT=\"y+=++%2811%2F6%29%286.4%29+-7%2F3\"   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=+9.4 ALIGN=MIDDLE  ALT=\"y=+9.4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "The coordinates are (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6.4 ALIGN=MIDDLE  ALT=\"6.4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9.4 ALIGN=MIDDLE  ALT=\"9.4\"   DISABLED_style= \"max-width:90%; height:auto;\" >).  This is the orthocenter.\r<BR>\n" );
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document.write( " \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0Acircle%28-3%2C4%2C.12%29%2C+locate%28-3%2C4%2CD%28-3%2C4%29%29%2C%0D%0Acircle%288%2C-2%2C.12%29%2C+locate%288%2C-2%2CE%288%2C-2%29%29%2C%0D%0Acircle%284%2C5%2C.12%29%2C+locate%284%2C5%2CF%284%2C5%29%29%2C%0D%0Aline%28-3%2C4%2C8%2C-2%29%2Cline%28-3%2C4%2C4%2C5%29%2Cline%288%2C-2%2C4%2C5%29%2C%0D%0Acircle%286.4%2C9.4%2C.12%29%2C+locate%286.4%2C9.4%2CO%286.4%2C9.4%29%29%2C%0D%0A+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%2811%2F6%29%28x%29+-7%2F3%2C%284%2F7%29+x+%2B40%2F7%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );