document.write( "Question 1091708: Find orthocentre of triangle with vertices (-2,-1),(6,-1),(2,5) \n" ); document.write( "

Algebra.Com's Answer #706163 by ikleyn(52802) $\"\"$ $\"About$

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\n" ); document.write( "Find orthocentre of triangle with vertices (-2,-1),(6,-1),(2,5)
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document.write( "The orthocenter is the point where all three altitudes of the triangle intersect. \r\n" );
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document.write( "Notice that the side of the triangle, connecting the vertices (-2,-1) and (6,-1), is horizontal line parallel to x-axis.\r\n" );
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document.write( "Therefore, the altitude drawn to this side is vertical line x = const, and since it passes through the point (2,5), this constant is equal to 2,\r\n" );
document.write( "and the equation of this altitude is x = 2.\r\n" );
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document.write( "We will find the orthocenter as the intersection point of this altitude with the other altitude drawn from the vertex (6,-1).\r\n" );
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document.write( "This altitude is perpendicular to the side of the triangle connecting two other points, (-2,-1) and (2,5).\r\n" );
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document.write( "The slope of this side/segment is m =  =  =  = .\r\n" );
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document.write( "Since the altitude from the point (6,-1) is perpendicular to this side, it has the slope .\r\n" );
document.write( "And since it passes through the point (6,-1), its equation is\r\n" );
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document.write( "y - (-1) = ,   or    \r\n" );
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document.write( "y + 1 = ,   or   y = .\r\n" );
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document.write( "Its intersection with the line x = 2 has y-coordinate  y =  =  .\r\n" );
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document.write( "Thus the orthocenter is the point (x,y) = (,).\r\n" );
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