document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=174559>Question 174559</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780030923456/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Hello..<BR>\n" );
document.write( "I need to find the orthocenter of a triangle with coordinates:<BR>\n" );
document.write( "G(-2,5)<BR>\n" );
document.write( "H(6,5)<BR>\n" );
document.write( "J(4,-1)<BR>\n" );
document.write( "AND...<BR>\n" );
document.write( "A(4,-3)<BR>\n" );
document.write( "B(8,5)<BR>\n" );
document.write( "C(8,-8)<BR>\n" );
document.write( "Thanks to whoever answers this question!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #129606 by </B><A HREF=/tutors/aboutme.mpl?userid=jojo14344><B>jojo14344(1513)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jojo14344><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=129606\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> </pre><font size=4><b><BR>\n" );
document.write( "Let's draw the triangle with following vertices:<BR>\n" );
document.write( "G(-2,5)<BR>\n" );
document.write( "H(6,5)<BR>\n" );
document.write( "J(4,-1)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-3%2C7%2C-2%2C6%2Cgraph%28400%2C400%2C-3%2C7%2C-2%2C6%29%2Ctriangle%28-2%2C5%2C6%2C5%2C4%2C-1%29%2Cblue%28locate%28-2.3%2C5%2CG%29%29%2Cblue%28locate%286.3%2C5%2CH%29%29%2Cblue%28locate%284%2C-1.3%2CJ%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now, we draw a line from the vertex that is perpendicular to the opposite side.<BR>\n" );
document.write( "1) from point G (-2,5) thru line HJ with points (6,5) & (4,-1):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BHJ%5D=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29=%28-1-5%29%2F%284-6%29=-6%2F-2 ALIGN=MIDDLE  ALT=\"m%5BHJ%5D=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29=%28-1-5%29%2F%284-6%29=-6%2F-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BHJ%5D=3 ALIGN=MIDDLE  ALT=\"m%5BHJ%5D=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, Slope<BR>\n" );
document.write( "Since perpendicular,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BG%5D=-1%2Fm%5BHJ%5D ALIGN=MIDDLE  ALT=\"m%5BG%5D=-1%2Fm%5BHJ%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BG%5D=-1%2F3 ALIGN=MIDDLE  ALT=\"m%5BG%5D=-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then,  via Slope-Intercept Form, <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;\" > on point G:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5=%28-1%2F3%29%28-2%29%2Bb ALIGN=MIDDLE  ALT=\"5=%28-1%2F3%29%28-2%29%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5=%282%2F3%29%2Bb ALIGN=MIDDLE  ALT=\"5=%282%2F3%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=b=5-%282%2F3%29=%2815-2%29%2F3=13%2F3 ALIGN=MIDDLE  ALT=\"b=5-%282%2F3%29=%2815-2%29%2F3=13%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Therefore, the line eqn -----><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=%28-1%2F3%29x%2B%2813%2F3%29%29 ALIGN=MIDDLE  ALT=\"highlight%28y=%28-1%2F3%29x%2B%2813%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , Line passing thru point G perpendicular to Line HJ<BR>\n" );
document.write( "2) from point H (6,5) thru line GJ with points (-2,5) & (4,-1):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BGJ%5D=%28-1-5%29%2F%284-%28-2%29%29=-6%2F%284%2B2%29=-6%2F6 ALIGN=MIDDLE  ALT=\"m%5BGJ%5D=%28-1-5%29%2F%284-%28-2%29%29=-6%2F%284%2B2%29=-6%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BGJ%5D=-1 ALIGN=MIDDLE  ALT=\"m%5BGJ%5D=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, Slope<BR>\n" );
document.write( "Also, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BH%5D=-1%2Fm%5BGJ%5D ALIGN=MIDDLE  ALT=\"m%5BH%5D=-1%2Fm%5BGJ%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >----><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BH%5D=-1%2F-1=1 ALIGN=MIDDLE  ALT=\"m%5BH%5D=-1%2F-1=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then, via point (6,5):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5=1%286%29%2Bb ALIGN=MIDDLE  ALT=\"5=1%286%29%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >----><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=5-6=-1 ALIGN=MIDDLE  ALT=\"b=5-6=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Therefore,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=1x-1%29 ALIGN=MIDDLE  ALT=\"highlight%28y=1x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, Line passing thru point H perpendicular to Line GJ<BR>\n" );
document.write( ".<BR>\n" );
document.write( "3)from point J (4,-1) thru Line GH with points (-2,5) & (6,5):<BR>\n" );
document.write( "We can see Line GH has no Slope, only horizontal line (y=5), so line perpendicular to it from point (4,-1) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B0%28y%29=4 ALIGN=MIDDLE  ALT=\"x%2B0%28y%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=4%29 ALIGN=MIDDLE  ALT=\"highlight%28x=4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > vertical line,slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinite ALIGN=MIDDLE  ALT=\"infinite\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Plug in \"highlighted\" in the graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-3%2C7%2C-2%2C6%2Cgraph%28400%2C400%2C-3%2C7%2C-2%2C6%2C%28-1%2F3%29x%2B%2813%2F3%29%2Cx-1%29%2Ctriangle%28-2%2C5%2C6%2C5%2C4%2C-1%29%2Cblue%28locate%28-2.3%2C5%2CA%29%29%2Cblue%28locate%286.3%2C5%2CH%29%29%2Cblue%28locate%284%2C-1.3%2CJ%29%29%2Cblue%28line%284%2C-1%2C4%2C5%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >----------><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-3%2C7%2C-2%2C6%2Cgrid%281%29%2Cgraph%28400%2C400%2C-3%2C7%2C-2%2C6%2C%28-1%2F3%29x%2B%2813%2F3%29%2Cx-1%29%2Ctriangle%28-2%2C5%2C6%2C5%2C4%2C-1%29%2Cblue%28locate%28-2.3%2C5%2CA%29%29%2Cblue%28locate%286.3%2C5%2CH%29%29%2Cblue%28locate%284%2C-1.3%2CJ%29%29%2Cblue%28line%284%2C-1%2C4%2C5%29%29%2Cred%28circle%284%2C3%2C.06%29%29%2Cred%28circle%284%2C3%2C.08%29%29%2Cred%28circle%284%2C3%2C.10%29%29%2Cred%28circle%284%2C3%2C.12%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Graph on the LEFT, See LINES from one vertex perpendicular to the other side.<BR>\n" );
document.write( "Graph on the RIGHT, Lines intersect at one point --->(4,3), orthocenter, ANSWER.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "NEXT<BR>\n" );
document.write( "Let's draw the Triangle with following vertices:<BR>\n" );
document.write( "Points<BR>\n" );
document.write( "A(4,-3)<BR>\n" );
document.write( "B(8,5)<BR>\n" );
document.write( "C(8,-8)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-1%2C9%2C-9%2C6%2Cgraph%28400%2C400%2C-1%2C9%2C-9%2C6%29%2Ctriangle%284%2C-3%2C8%2C5%2C8%2C-8%29%2Cblue%28locate%283.7%2C-3%2CA%29%29%2Cblue%28locate%288.2%2C5.2%2CB%29%29%2Cblue%28locate%288.2%2C-8.2%2CC%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The same we did the last one,drawing a Line from the vertex that is perpendicular to the opposite side:<BR>\n" );
document.write( "1)As you can see on point A (4,-3) is going thru vertical LINE BC and the line from point (4,-3) should be perpendicular to BC----><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=-3%29 ALIGN=MIDDLE  ALT=\"highlight%28y=-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "2)Point B (8,5) thru Line AC with points (4,-3) & (8,-8):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BAC%5D=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29=%28-8-%28-3%29%29%2F%288-4%29=%28-8%2B3%29%2F4 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >}<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BAC%5D=-5%2F4 ALIGN=MIDDLE  ALT=\"m%5BAC%5D=-5%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BB%5D=-1%2Fm%5BAC%5D ALIGN=MIDDLE  ALT=\"m%5BB%5D=-1%2Fm%5BAC%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, perpendicular right?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BB%5D=-1%2F%28-5%2F4%29=4%2F5 ALIGN=MIDDLE  ALT=\"m%5BB%5D=-1%2F%28-5%2F4%29=4%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then,thru point (8,5): via Slope-Intercept Form <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;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5=%284%2F5%298%2Bb ALIGN=MIDDLE  ALT=\"5=%284%2F5%298%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5=32%2F5%2Bb ALIGN=MIDDLE  ALT=\"5=32%2F5%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=5-%2832%2F5%29=%2825-32%29%2F5=-7%2F5 ALIGN=MIDDLE  ALT=\"b=5-%2832%2F5%29=%2825-32%29%2F5=-7%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, it follows -------><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=%284%2F5%29x-%287%2F5%29%29 ALIGN=MIDDLE  ALT=\"highlight%28y=%284%2F5%29x-%287%2F5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "3)Point C (8,-8) thru line AB with points (4,-3) & (8,5):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BAB%5D=%285-%28-3%29%29%2F%288-4%29=%285%2B3%29%2F4 ALIGN=MIDDLE  ALT=\"m%5BAB%5D=%285-%28-3%29%29%2F%288-4%29=%285%2B3%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BAB%5D=8%2F4=2 ALIGN=MIDDLE  ALT=\"m%5BAB%5D=8%2F4=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5BC%5D=-1%2Fm%5BAB%5D=-1%2F2 ALIGN=MIDDLE  ALT=\"m%5BC%5D=-1%2Fm%5BAB%5D=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Thru point (8,-8):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8=%28-1%2Fcross%282%291%29%2A%28cross%288%294%29%2Bb ALIGN=MIDDLE  ALT=\"-8=%28-1%2Fcross%282%291%29%2A%28cross%288%294%29%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >-----><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8=-4%2Bb ALIGN=MIDDLE  ALT=\"-8=-4%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=-8%2B4=-4 ALIGN=MIDDLE  ALT=\"b=-8%2B4=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "It follows------><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=%28-1%2F2%29%28x%29-4%29 ALIGN=MIDDLE  ALT=\"highlight%28y=%28-1%2F2%29%28x%29-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Plug in \"highlighted\" in the graph;<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-6%2C9%2C-9%2C6%2Cgraph%28400%2C400%2C-6%2C9%2C-9%2C6%2C-3%2C%284%2F5%29x-%287%2F5%29%2C%28-1%2F2%29x-4%29%2Ctriangle%284%2C-3%2C8%2C5%2C8%2C-8%29%2Cblue%28locate%283.7%2C-3%2CA%29%29%2Cblue%28locate%288.2%2C5.2%2CB%29%29%2Cblue%28locate%288.2%2C-8.2%2CC%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >------><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-6%2C9%2C-9%2C6%2Cgrid%281%29%2Cgraph%28400%2C400%2C-6%2C9%2C-9%2C6%2C-3%2C%284%2F5%29x-%287%2F5%29%2C%28-1%2F2%29x-4%29%2Ctriangle%284%2C-3%2C8%2C5%2C8%2C-8%29%2Cblue%28locate%283.7%2C-3%2CA%29%29%2Cblue%28locate%288.2%2C5.2%2CB%29%29%2Cblue%28locate%288.2%2C-8.2%2CC%29%29%2Cred%28circle%28-2%2C-3%2C.06%29%29%2Cred%28circle%28-2%2C-3%2C.08%29%29%2Cred%28circle%28-2%2C-3%2C.10%29%29%2Cred%28circle%28-2%2C-3%2C0%2C0.12%29%29%2Cred%28circle%28-2%2C-3%2C0.16%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "-------> Orthocenter lies outside the Triangle, being \"Obtuse Triangle\" >>>> point (-2,-3), ANSWER<BR>\n" );
document.write( "Thank you,<BR>\n" );
document.write( "Jojo</pre>\r<BR>\n" );
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