document.write( "Question 823740: Find the orthocenter of triangle KLM with vertices K(2,-2), L(4,6), M(8,-2)
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Algebra.Com's Answer #495762 by Edwin McCravy(20056)\"\" \"About 
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document.write( "The orthocenter is the point where the three (green)\r\n" );
document.write( "altitudes intersect.  Altitudes are perpendicular to the\r\n" );
document.write( "sides.  \r\n" );
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document.write( "We will find the equation of altitude AM and find\r\n" );
document.write( "where it intersects the altitude LC\r\n" );
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document.write( "First we must find the slope of AM.  Since AM is\r\n" );
document.write( "perpendicular to KL, we will find the slope of KL and\r\n" );
document.write( "take its negative reciprocal:\r\n" );
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document.write( "We use the slope formula to find the slope of KL.\r\n" );
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document.write( "m = \"%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\"\r\n" );
document.write( "where (x1,y1) = K(2,-2)\r\n" );
document.write( "and where (x2,y2) = L(4,6)\r\n" );
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document.write( "m = \"%286-%28-2%29%29%2F%284-2%29\"\"%22%22=%22%22\"\"%286%2B2%29%2F2\"\"%22%22=%22%22\"\"8%2F2\" = 4.\r\n" );
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document.write( "Therefore the slope of MA is the negative reciprocal of\r\n" );
document.write( "4, which is \"-1%2F4\".\r\n" );
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document.write( "To find the equation of MA we use the point-slope formula:\r\n" );
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document.write( "y - y1 = m(x - x1)\r\n" );
document.write( "where (x1,y1) = M(8,-2)\r\n" );
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document.write( "y - (-2) = \"-1%2F4\"(x - 8)\r\n" );
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document.write( "   y + 2 = \"-1%2F4\"x + 2\r\n" );
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document.write( "       y = \"-1%2F4\"x\r\n" );
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document.write( "The altitude LC is vertical (since KM is horizontal)\r\n" );
document.write( "So the equation of LC is x = 4\r\n" );
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document.write( "So we substitute x = 4 in y = \"-1%2F4\"x and get\r\n" );
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document.write( "y = \"-1%2F4\"(4) = -1.\r\n" );
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document.write( "So the orthocenter O is (4,-1)\r\n" );
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document.write( "--------------------------------\r\n" );
document.write( "Note:\r\n" );
document.write( "We could have used the altitude KB instead of MA.\r\n" );
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document.write( "Edwin
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