document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=127639>Question 127639</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. An altitude of a triangle is a line segment from a vertex perpendicular to the opposite side. Find the equations of the altitudes of the triangle with vertices (4, 5),(-4, 1) and (2, -5). Do this by solving a system of two of two of the altitude equations and showing that the intersection point also belongs to the third line. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #93523 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=93523\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If we plot the points and connect them, we get this triangle:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0Agraph%28500%2C500%2C-10%2C10%2C-10%2C10%2C0%29%2C%0D%0Acircle%284%2C5%2C0.08%29%2C%0D%0Acircle%284%2C5%2C0.1%29%2C%0D%0Acircle%284%2C5%2C0.12%29%2C%0D%0Acircle%28-4%2C1%2C0.08%29%2C%0D%0Acircle%28-4%2C1%2C0.1%29%2C%0D%0Acircle%28-4%2C1%2C0.12%29%2C%0D%0Acircle%282%2C-5%2C0.08%29%2C%0D%0Acircle%282%2C-5%2C0.1%29%2C%0D%0Acircle%282%2C-5%2C0.12%29%2C%0D%0Aline%284%2C5%2C-4%2C1%29%2C%0D%0Aline%28-4%2C1%2C2%2C-5%29%2C%0D%0Aline%282%2C-5%2C4%2C5%29%2C%0D%0Alocate%284%2C5%2CA%29%2C%0D%0Alocate%28-4.2%2C1%2CB%29%2C%0D%0Alocate%282%2C-5%2CC%29%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let point <BR>\n" );
document.write( "A=(xA,yA)<BR>\n" );
document.write( "B=(xB,yB)<BR>\n" );
document.write( "C=(xC,yC)\r<BR>\n" );
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document.write( "Let's find the equation of the segment AB\r<BR>\n" );
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document.write( "Start with the general formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28%28yA-yB%29%2F%28xA-xB%29%29%28x-xA%29%2ByA ALIGN=MIDDLE  ALT=\"y=%28%28yA-yB%29%2F%28xA-xB%29%29%28x-xA%29%2ByA\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Plug in the given points\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28%28+5-+1%29%2F%284%2B4%29%29%28x-4%29%2B+5 ALIGN=MIDDLE  ALT=\"y=%28%28+5-+1%29%2F%284%2B4%29%29%28x-4%29%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Simplify and combine like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29x%2B3 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the equation of the line through AB is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29x%2B3 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's find the equation of the segment BC\r<BR>\n" );
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document.write( "Start with the general formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28%28yB-yC%29%2F%28xB-xC%29%29%28x-xB%29%2ByB ALIGN=MIDDLE  ALT=\"y=%28%28yB-yC%29%2F%28xB-xC%29%29%28x-xB%29%2ByB\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Plug in the given points\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28%28+1%2B5%29%2F%28-4-2%29%29%28x%2B4%29%2B+1 ALIGN=MIDDLE  ALT=\"y=%28%28+1%2B5%29%2F%28-4-2%29%29%28x%2B4%29%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Simplify and combine like terms\r<BR>\n" );
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document.write( "<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" );
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document.write( "So the equation of the line through BC is <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" );
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document.write( "-------------------------------\r<BR>\n" );
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document.write( "Let's find the equation of the segment CA\r<BR>\n" );
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document.write( "Start with the general formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28%28yC-yA%29%2F%28xC-xA%29%29%28x-xC%29%2ByC ALIGN=MIDDLE  ALT=\"y=%28%28yC-yA%29%2F%28xC-xA%29%29%28x-xC%29%2ByC\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Plug in the given points\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28%28+-5-+5%29%2F%282-4%29%29%28x-2%29%2B+-5 ALIGN=MIDDLE  ALT=\"y=%28%28+-5-+5%29%2F%282-4%29%29%28x-2%29%2B+-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Simplify and combine like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5x-15 ALIGN=MIDDLE  ALT=\"y=5x-15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the equation of the line through CA is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5x-15 ALIGN=MIDDLE  ALT=\"y=5x-15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So we have these equations of the lines that make up the triangle\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0Agraph%28500%2C500%2C-10%2C10%2C-10%2C10%2C0%29%2C%0D%0Acircle%284%2C5%2C0.08%29%2C%0D%0Acircle%284%2C5%2C0.1%29%2C%0D%0Acircle%284%2C5%2C0.12%29%2C%0D%0Acircle%28-4%2C1%2C0.08%29%2C%0D%0Acircle%28-4%2C1%2C0.1%29%2C%0D%0Acircle%28-4%2C1%2C0.12%29%2C%0D%0Acircle%282%2C-5%2C0.08%29%2C%0D%0Acircle%282%2C-5%2C0.1%29%2C%0D%0Acircle%282%2C-5%2C0.12%29%2C%0D%0Aline%284%2C5%2C-4%2C1%29%2C%0D%0Aline%28-4%2C1%2C2%2C-5%29%2C%0D%0Aline%282%2C-5%2C4%2C5%29%2C%0D%0Alocate%284%2C5%2CA%29%2C%0D%0Alocate%28-4.2%2C1%2CB%29%2C%0D%0Alocate%282%2C-5%2CC%29%2C%0D%0Alocate%28-4.1%2C4%2Cy=%281%2F2%29x%2B3%29%2C%0D%0Alocate%283.5%2C1%2Cy=5x-15%29%0D%0Alocate%28-3.9%2C-2.2%2Cy=-x-3%29%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So to find the equation of the line that is perpendicular to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29x%2B3 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > that goes through the point C(2,-5), simply negate and invert the slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2F1=-2 ALIGN=MIDDLE  ALT=\"-2%2F1=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now plug the slope and the point (2,-5) into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-y%5B1%5D=m%28x-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"y-y%5B1%5D=m%28x-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-%28-5%29=-2%28x-2%29 ALIGN=MIDDLE  ALT=\"y-%28-5%29=-2%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x-1 ALIGN=MIDDLE  ALT=\"y=-2x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve for y and simplify\r<BR>\n" );
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document.write( "So the altitude for vertex C is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x-1 ALIGN=MIDDLE  ALT=\"y=-2x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now to find the equation of the line that is perpendicular to <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;\" > that goes through the point A(4,5), simply negate and invert the slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F1 ALIGN=MIDDLE  ALT=\"-1%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28-1%2F1%29=1 ALIGN=MIDDLE  ALT=\"-%28-1%2F1%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now plug the slope and the point (2,-5) into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-y%5B1%5D=m%28x-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"y-y%5B1%5D=m%28x-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-5=1%28x-4%29 ALIGN=MIDDLE  ALT=\"y-5=1%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%2B1 ALIGN=MIDDLE  ALT=\"y=x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve for y and simplify\r<BR>\n" );
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document.write( "So the altitude for vertex A is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%2B1 ALIGN=MIDDLE  ALT=\"y=x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now to find the equation of the line that is perpendicular to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5x-15 ALIGN=MIDDLE  ALT=\"y=5x-15\"   DISABLED_style= \"max-width:90%; height:auto;\" > that goes through the point B(-4,1), simply negate and invert the slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F1 ALIGN=MIDDLE  ALT=\"5%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%281%2F5%29=-1%2F5 ALIGN=MIDDLE  ALT=\"-%281%2F5%29=-1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now plug the slope and the point (-4,1) into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-y%5B1%5D=m%28x-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"y-y%5B1%5D=m%28x-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-1=-%281%2F5%29%28x%2B4%29 ALIGN=MIDDLE  ALT=\"y-1=-%281%2F5%29%28x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%281%2F5%29x%2B1%2F5 ALIGN=MIDDLE  ALT=\"y=-%281%2F5%29x%2B1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve for y and simplify\r<BR>\n" );
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document.write( "So the altitude for vertex B is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%281%2F5%29x%2B1%2F5 ALIGN=MIDDLE  ALT=\"y=-%281%2F5%29x%2B1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's solve the system \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x-1 ALIGN=MIDDLE  ALT=\"y=-2x-1\"   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%2B1 ALIGN=MIDDLE  ALT=\"y=x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1=-2x-1 ALIGN=MIDDLE  ALT=\"x%2B1=-2x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%2B1 ALIGN=MIDDLE  ALT=\"y=x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the first equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x=-2 ALIGN=MIDDLE  ALT=\"3x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 2x to both sides and subtract 2 from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2%2F3 ALIGN=MIDDLE  ALT=\"x=-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 3 to isolate x\r<BR>\n" );
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document.write( "Now plug this into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%2B1 ALIGN=MIDDLE  ALT=\"y=x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2%2F3%2B1 ALIGN=MIDDLE  ALT=\"y=-2%2F3%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2F3 ALIGN=MIDDLE  ALT=\"y=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the orthocenter is (-2/3,1/3)\r<BR>\n" );
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document.write( "So if we plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2%2F3 ALIGN=MIDDLE  ALT=\"x=-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the third equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%281%2F5%29x%2B1%2F5 ALIGN=MIDDLE  ALT=\"y=-%281%2F5%29x%2B1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%281%2F5%29%28-2%2F3%29%2B1%2F5 ALIGN=MIDDLE  ALT=\"y=-%281%2F5%29%28-2%2F3%29%2B1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%2F15%2B1%2F5 ALIGN=MIDDLE  ALT=\"y=2%2F15%2B1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3%2F15 ALIGN=MIDDLE  ALT=\"y=3%2F15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2F3 ALIGN=MIDDLE  ALT=\"y=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the orthocenter lies on the third altitude\r<BR>\n" );
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document.write( "-----------------------------------\r<BR>\n" );
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document.write( "Summary:\r<BR>\n" );
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document.write( "So the equations of the altidudes are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x-1 ALIGN=MIDDLE  ALT=\"y=-2x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%2B1 ALIGN=MIDDLE  ALT=\"y=x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%281%2F5%29%2B1%2F5 ALIGN=MIDDLE  ALT=\"y=-%281%2F5%29%2B1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );