document.write( "Question 1060786: Hi! I'm wondering how to solve for the orthocenter. Problem: Find the orthocenter of triangle JKL with vertices J(2,1), K(9,1), and L(4,6). \n" ); document.write( "

Algebra.Com's Answer #675721 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Hi! I'm wondering how to solve for the orthocenter. Problem: Find the orthocenter of triangle JKL with vertices J(2,1), K(9,1), and L(4,6).
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document.write( "1.  Orthocenter is the common intersection point of altitudes of a triangle.\r\n" );
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document.write( "    (In any triangle, the three altitudes are concurrent and intersect in one point.)\r\n" );
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document.write( "    So all you need to do is to find the intersection of (any) two altitudes of the given triangle.\r\n" );
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document.write( "2.  One side of the triangle, the side JK, is horizontal line y = 1 in the coordinate plane, parallel to x-axis.\r\n" );
document.write( "    Hence, the altitude to this side is a VERTICAL line parallel to y-axis. \r\n" );
document.write( "    Since this altitude passes through the point L=(4,6), the equation of this altitude is \r\n" );
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document.write( "    x = 4.             (1)\r\n" );
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document.write( "3.  The side KL of the triangle has the slope m =  =  = -1.\r\n" );
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document.write( "    Hence, the altitude of the triangle drawn to this side, has the slope  = 1 and, therefore,\r\n" );
document.write( "    has an equation \r\n" );
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document.write( "    y = x+b.           (2)\r\n" );
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document.write( "    Since this altitude passes through the point J(2,1), from the equation (2) we have 1 = 2 + b.\r\n" );
document.write( "    Hence, b =  = -1 and finally, the equation (2) is\r\n" );
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document.write( "    y = x-1.           (3)\r\n" );
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document.write( "4.  Since we are looking for the intersection of the straight lines (1) and (3), we must solve these equations as a system.\r\n" );
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document.write( "    Then you have \r\n" );
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document.write( "    y = 3.\r\n" );
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document.write( "Answer.  The orthocenter is the point  (x,y) = (4,3).\r\n" );
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