document.write( "Question 379853: Here are the coordinates to a triangle A(0,0) B(12,6) C(18,0)
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Algebra.Com's Answer #269681 by Edwin McCravy(20059) $\"\"$ $\"About$

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document.write( "We draw the triangle:\r\n" );
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document.write( "The orthocenter is where all three of the extended altitudes intersect.\r\n" );
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document.write( "We draw the altitude from B to AC\r\n" );
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document.write( "Since it's vertical and goes through B(12,6) its equation is \r\n" );
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document.write( "x = 12\r\n" );
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document.write( "Now we draw another altitude from A to BC. but to do that we have to first\r\n" );
document.write( "extend BC.\r\n" );
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document.write( "To find the equation of this second (red) altitude, we need to know its slope.\r\n" );
document.write( "It is perpendicular to BC.  So first we find the slope of BC:\r\n" );
document.write( "A(0,0) B(12,6) C(18,0)\r\n" );
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document.write( "So the slope of the red altitude is the opposite-signed reciprocal of\r\n" );
document.write( "the slope of BC.  The opposite signed reciprocal of -1 is .\r\n" );
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document.write( "Since it goes through the origin, its y-intercept is b=0 and so its\r\n" );
document.write( "equation is \r\n" );
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document.write( "y = 1x + 0 \r\n" );
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document.write( "y = x\r\n" );
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document.write( "To show where the green altitude intersects the red altitude,\r\n" );
document.write( "we must extend both of them:\r\n" );
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document.write( "We just need to find the point where the red altitude and the \r\n" );
document.write( "green altitude intersect, for that is the orthocenter.\r\n" );
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document.write( "We solve the easy system:\r\n" );
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document.write( "That has the solution (12,12),\r\n" );
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document.write( "so the orthocenter is O(12,12)\r\n" );
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document.write( "We didn't need the third altitude. If we were to draw it we \r\n" );
document.write( "would find that it would also go through the orthocenter:\r\n" );
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document.write( "Edwin

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