document.write( "Question 721940: how do i prove an altitude? What are the steps to doing this in a coordinate geometry question where they ask you to prove that line CD in triangle ABC is an altitude and is bisecting the base of the triangle. \n" ); document.write( "
Algebra.Com's Answer #442573 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "You don't need the Pythagorean theorem.\r\n" );
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document.write( "Given ߡABC with vertices A(-1,2), B(5,-4), C(9,6), prove that CD\r\n" );
document.write( "is both an altitude (perpendicular to base AB) and a median\r\n" );
document.write( "(bisects the base) where D is the point D(-4,3).\r\n" );
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document.write( "Draw ߡABC and AD\r\n" );
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document.write( "We need to show that \r\n" );
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document.write( "(1) CD ⊥ AB  and  (2) D is the midpoint of AB.\r\n" );
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document.write( "For (1), we find the slopes of CD and AB and show that they are negative\r\n" );
document.write( "reciprocals, that their product is -1.\r\n" );
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document.write( "Slope formula:\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" );
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document.write( "To find the slope of CD:\r\n" );
document.write( "where (x1,y1) = C(6,8)\r\n" );
document.write( "and where (x2,y2) = D(-4,3)\r\n" );
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document.write( "m = \"%28%283%29-%288%29%29%2F%28%28-4%29-%286%29%29\" = \"-5%2F-10\" = \"1%2F2\"\r\n" );
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document.write( "To find the slope of AB:\r\n" );
document.write( "where (x1,y1) = A(-6,7)\r\n" );
document.write( "and where (x2,y2) = B(-2,-1)\r\n" );
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document.write( "m = \"%28%28-1%29-%287%29%29%2F%28%28-2%29-%28-6%29%29\" = \"-8%2F%28-2%2B6%29\" = \"%28-8%29%2F4\" = -2\r\n" );
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document.write( "Since \"1%2F2\" and \"-2\" are negative reciprocals,\r\n" );
document.write( "their product is -1, we have proved that CD ⊥ AB,\r\n" );
document.write( "which proves that CD is an altitude to base AB.\r\n" );
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document.write( "For (2), we use the midpoint formula to show that D(-4,3) is the\r\n" );
document.write( "midpoint of AB.\r\n" );
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document.write( "Midpoint formula:\r\n" );
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document.write( "Midpoint = \r\n" );
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document.write( "where (x1,y1) = A(-6,7)\r\n" );
document.write( "and where (x2,y2) = B(-2,-1)\r\n" );
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document.write( "Midpoint =  = \"%28matrix%281%2C3%2C%28-8%29%2F2%2C+%22%2C%22%2C%286%29%2F2%29%29\" = (-4,3) which is point D.\r\n" );
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document.write( "Thus CD is also a median to the base AB of ߡABC.\r\n" );
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document.write( "Edwin
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