document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181232>Question 1181232</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given triangle ABC with vertices A(6, 3), B(-5, -8), and C(-3, 6), let BP and CQ be the perpendiculars dropped from B and C to their opposite sides of triangle ABC. Find the coordinates of orthocenter H of triangle ABC. \r<BR>\n" );
document.write( "\n" );
document.write( "P.S. Can you show full solution, thanks.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #811161 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</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=811161\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABC ALIGN=MIDDLE  ALT=\"ABC\"   DISABLED_style= \"max-width:90%; height:auto;\" > with vertices \r<BR>\n" );
document.write( "\n" );
document.write( "A(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   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;\" >), <BR>\n" );
document.write( "B(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   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;\" >), and <BR>\n" );
document.write( "C(<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=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >), \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "let BP and CQ be the perpendiculars dropped from B and C to their opposite sides of triangle ABC\r<BR>\n" );
document.write( "\n" );
document.write( "a line through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BP+ ALIGN=MIDDLE  ALT=\"BP+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is perpendicular to a line through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC ALIGN=MIDDLE  ALT=\"AC\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "slope of a line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx%2Bb ALIGN=MIDDLE  ALT=\"y=mx%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC ALIGN=MIDDLE  ALT=\"AC\"   DISABLED_style= \"max-width:90%; height:auto;\" > is\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%286-3%29%2F%28-3-6%29=-3%2F9=-1%2F3 ALIGN=MIDDLE  ALT=\"m=%286-3%29%2F%28-3-6%29=-3%2F9=-1%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=-%281%2F3%29x%2Bb ALIGN=MIDDLE  ALT=\"y=-%281%2F3%29x%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >.......use one point to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"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=3=-%281%2F3%296%2Bb ALIGN=MIDDLE  ALT=\"3=-%281%2F3%296%2Bb\"   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=3=-2%2Bb ALIGN=MIDDLE  ALT=\"3=-2%2Bb\"   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 ALIGN=MIDDLE  ALT=\"b=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "a line through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC ALIGN=MIDDLE  ALT=\"AC\"   DISABLED_style= \"max-width:90%; height:auto;\" > is:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%281%2F3%29x%2B5 ALIGN=MIDDLE  ALT=\"y=-%281%2F3%29x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "then a perpendicular line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BP+ ALIGN=MIDDLE  ALT=\"BP+\"   DISABLED_style= \"max-width:90%; height:auto;\" > will have a slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F%28-1%2F3%29=3 ALIGN=MIDDLE  ALT=\"-1%2F%28-1%2F3%29=3\"   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=3x%2Bb ALIGN=MIDDLE  ALT=\"y=3x%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >........use point B\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8=3%28-5%29%2Bb ALIGN=MIDDLE  ALT=\"-8=3%28-5%29%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8=-15%2Bb ALIGN=MIDDLE  ALT=\"-8=-15%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15-8=b ALIGN=MIDDLE  ALT=\"15-8=b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=7 ALIGN=MIDDLE  ALT=\"b=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "and a perpendicular line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BP+ ALIGN=MIDDLE  ALT=\"BP+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%2B7 ALIGN=MIDDLE  ALT=\"y=3x%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "now we need a line that passes through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CQ ALIGN=MIDDLE  ALT=\"CQ\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is perpendicular to BA\r<BR>\n" );
document.write( "\n" );
document.write( "line BA have a slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%28-8-3%29%2F%28-5-6%29=-11%2F-11=1 ALIGN=MIDDLE  ALT=\"m=%28-8-3%29%2F%28-5-6%29=-11%2F-11=1\"   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=1x%2Bb ALIGN=MIDDLE  ALT=\"y=1x%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >......use one point to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"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=3=1%2A6%2Bb ALIGN=MIDDLE  ALT=\"3=1%2A6%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3-6=b ALIGN=MIDDLE  ALT=\"3-6=b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-3 ALIGN=MIDDLE  ALT=\"b=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "a line that passes through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BA ALIGN=MIDDLE  ALT=\"BA\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x-3 ALIGN=MIDDLE  ALT=\"y=x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >  \r<BR>\n" );
document.write( "\n" );
document.write( "a line that passes through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CQ ALIGN=MIDDLE  ALT=\"CQ\"   DISABLED_style= \"max-width:90%; height:auto;\" > will have a slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F1=-1 ALIGN=MIDDLE  ALT=\"-1%2F1=-1\"   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=-x%2Bb ALIGN=MIDDLE  ALT=\"y=-x%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >....use coordinates of C<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6=-%28-3%29%2Bb ALIGN=MIDDLE  ALT=\"6=-%28-3%29%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6=3%2Bb ALIGN=MIDDLE  ALT=\"6=3%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=3 ALIGN=MIDDLE  ALT=\"b=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "a line that passes through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CQ ALIGN=MIDDLE  ALT=\"CQ\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-x%2B3 ALIGN=MIDDLE  ALT=\"y=-x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the orthocenter <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H ALIGN=MIDDLE  ALT=\"H\"   DISABLED_style= \"max-width:90%; height:auto;\" > is intersection point of a lines that passes through  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BP+ ALIGN=MIDDLE  ALT=\"BP+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CQ ALIGN=MIDDLE  ALT=\"CQ\"   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=3x%2B7 ALIGN=MIDDLE  ALT=\"y=3x%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-x%2B3 ALIGN=MIDDLE  ALT=\"y=-x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--------------------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B7=-x%2B3 ALIGN=MIDDLE  ALT=\"3x%2B7=-x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2Bx=3-7 ALIGN=MIDDLE  ALT=\"3x%2Bx=3-7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x=-4 ALIGN=MIDDLE  ALT=\"4x=-4\"   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 ALIGN=MIDDLE  ALT=\"x=-1\"   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=3%28-1%29%2B7 ALIGN=MIDDLE  ALT=\"y=3%28-1%29%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-3%2B7 ALIGN=MIDDLE  ALT=\"y=-3%2B7\"   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 ALIGN=MIDDLE  ALT=\"y=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "  the coordinates of orthocenter H of triangle ABC are: (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   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;\" >)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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%286%2C3%2C.12%29%2C+locate%286%2C3%2CA%29%2C%0D%0Acircle%28-5%2C-8%2C.12%29%2C+locate%28-5%2C-8%2CB%29%2C%0D%0Acircle%28-3%2C6%2C.12%29%2C+locate%28-3%2C6%2CC%29%2C%0D%0Acircle%28-1%2C4%2C.12%29%2C+locate%28-1%2C4%2CH%29%2C%0D%0A%0D%0Agreen%28line%286%2C3%2C-5%2C-8%29%29%2C%0D%0Agreen%28line%286%2C3%2C-3%2C6%29%29%2C%0D%0Agreen%28line%28-3%2C6%2C-5%2C-8%29%29%2C%0D%0Agraph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C-x%2B3%2C-x%2B3%2C3x%2B7%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );