document.write( "Question 647850: Show that the points given are vertices of a right triangle: (-1,3), (3,5), (5,1)\r
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Algebra.Com's Answer #406442 by Edwin McCravy(20086)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "There are two ways to do this:\r\n" );
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document.write( "1. Find the lengths of all the sides by the distance formula and showing that\r\n" );
document.write( "they satisfy the Pythagorean theorem c²=a²+b²  \r\n" );
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document.write( "2. Find the slopes of the sides and observe that one of them is the \r\n" );
document.write( "\"negative reciprocal\" of another one.\r\n" );
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document.write( "The second way is easier.\r\n" );
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document.write( "If you make a sketch you can tell which two sides are the perpendicular\r\n" );
document.write( "ones:\r\n" );
document.write( "(-1,3), (3,5), (5,1)\r\n" );
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document.write( "It has to be the red ones.  So we don't need to bother finding the\r\n" );
document.write( "slope of the green line, (the hypotenuse).\r\n" );
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document.write( "Slope formula\r\n" );
document.write( "m = \"%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\" with (x1, y1) = (-1,3) and (x2, y2) = (3,5)\r\n" );
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document.write( "Substituting:\r\n" );
document.write( "For the red line on the left:\r\n" );
document.write( "m = \"%28%285%29-%283%29%29%2F%28%283%29-%28-1%29%29\" = \"%285-3%29%2F%283%2B1%29\" = \"2%2F4\" = \"1%2F2\"\r\n" );
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document.write( "m = \"%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\" with (x1, y1) = (-1,3) and (x2, y2) = (3,5)\r\n" );
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document.write( "Substituting:\r\n" );
document.write( "For the red line on the right:\r\n" );
document.write( "m = \"%28%281%29-%285%29%29%2F%28%285%29-%283%29%29\" = \"%281-5%29%2F%285-3%29\" = \"%28-4%29%2F2\" = \"-2\"\r\n" );
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document.write( "And since those slopes \"1%2F2\" and \"-2\" are \"negative reciprocals\",\r\n" );
document.write( "those lines with those slopes are perpendicular.  [You can tell that they are\r\n" );
document.write( "negative reciprocals either because you observe that one is the reciprocal of\r\n" );
document.write( "the other one with the sign changed or because you observe that if you multiply\r\n" );
document.write( "them you will get -1, which is true of all fractions multiplied by their\r\n" );
document.write( " negative reciprocals.\r\n" );
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document.write( "Edwin
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