document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1103165>Question 1103165</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the orthocenter of a triangle with the given vertices: <BR>\n" );
document.write( "U(-4,-9), V(-4,6),W(5,-3) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #717847 by </B><A HREF=/tutors/aboutme.mpl?userid=richwmiller><B>richwmiller(17219)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richwmiller><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=richwmiller><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=717847\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In our example, we will use the following coordinates as the vertices of the triangle. A (3, 1)  B(2, 2) C (3, 5) <BR>\n" );
document.write( "Find the equations of 2 segments of the triangle (for our example we will find the equations for AB, and BC)<BR>\n" );
document.write( "Once you have the equations from step #1, you can find the slope of the corresponding perpendicular lines.<BR>\n" );
document.write( "You will use the slopes you have found from step #2, and the corresponding opposite vertex to find the equations of the 2 lines.<BR>\n" );
document.write( "Once you have the equation of the 2 lines from step #3, you can solve the corresponding x and y, which is the coordinates of the orthocenter.<BR>\n" );
document.write( "The steps might seem daunting, but once you actually work through the problem, you will see that it is a very easy process.\r<BR>\n" );
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document.write( "Step 1:  Find equations of the line segments AB and BC.\r<BR>\n" );
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document.write( "To find any line segment, you will need to find the slope of the line and then the corresponding y-intercept.<BR>\n" );
document.write( "A (3, 1)  B(2, 2)  C (3, 5)<BR>\n" );
document.write( "Slope of AB = (1-2)/(3-2) = -1/1 = -1<BR>\n" );
document.write( "y = mx + b (substitute m = -1, x = 3, y = 1)<BR>\n" );
document.write( "1 = -1(3) + b<BR>\n" );
document.write( "b = 4<BR>\n" );
document.write( "Equation of AB:  y = -1x + 4\r<BR>\n" );
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document.write( "Slope of BC = (2-5)/(2-3) = -3/-1 = 3<BR>\n" );
document.write( "y = mx + b (substitute m = 3, x = 2, y = 2)<BR>\n" );
document.write( "2 = 3(2) + b<BR>\n" );
document.write( "b = -4<BR>\n" );
document.write( "Equation of BC:  y = 3x - 4<BR>\n" );
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document.write( "Step 2:  Find the slope of the corresponding perpendicular lines\r<BR>\n" );
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document.write( "Slope of AB = -1<BR>\n" );
document.write( "Slope of perpendicular line to AB:  -1*m = -1 -> m = 1\r<BR>\n" );
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document.write( "Slope of BC = 3<BR>\n" );
document.write( "Slope of perpendicular line to BC:  3*m = -1 -> m = -1/3\r<BR>\n" );
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document.write( "Step 3:  Find the equation of the perpendicular lines\r<BR>\n" );
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document.write( "Slope of perpendicular line to AB:  m = 1<BR>\n" );
document.write( "We will use the coordinate of the opposite vertex (point C) to find the equation of the line.\r<BR>\n" );
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document.write( "y = mx + b (substitute m = 1, x = 3, y = 5)<BR>\n" );
document.write( "5 = 1(3) + b<BR>\n" );
document.write( "b = 2<BR>\n" );
document.write( "Equation of perpendicular line to AB:  y = 1x + 2\r<BR>\n" );
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document.write( "Slope of perpendicular line to BC:  m = -1/3<BR>\n" );
document.write( "We will use the coordinate of the opposite vertex (point A) to find the equation of the line.\r<BR>\n" );
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document.write( "y = mx + b (substitute m = -1/3, x = 3, y = 1)<BR>\n" );
document.write( "1 = -1/3*(3) + b<BR>\n" );
document.write( "b = 2<BR>\n" );
document.write( "Equation of perpendicular line to AB:  y = -1/3x + 2\r<BR>\n" );
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document.write( "Step 4:  solve 2 perpendicular lines\r<BR>\n" );
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document.write( "equation 1:  y = 1x + 2<BR>\n" );
document.write( "equation 2:  y = -1/3x + 2\r<BR>\n" );
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document.write( "Solving for x and y:\r<BR>\n" );
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document.write( "1x + 2 = -1/3x + 2<BR>\n" );
document.write( "4/3x = 0<BR>\n" );
document.write( "x = 0\r<BR>\n" );
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document.write( "y = 1(0) + 2<BR>\n" );
document.write( "y = 2\r<BR>\n" );
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document.write( "The coordinates are (0, 2).  This is the orthocenter. </SPAN>\n" );
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