document.write( "Question 1091801: One leg of a right triangle has a length of 11 and all sides are whole numbers. Find the lengths of the other two sides and the hypotenuse? \n" ); document.write( "

Algebra.Com's Answer #706368 by ikleyn(52794) $\"\"$ $\"About$

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\n" ); document.write( "One leg of a right triangle has a length of 11 and all sides are whole numbers. Find the lengths of the other two sides and the hypotenuse?
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\n" ); document.write( "\n" ); document.write( "I read the solution by @jorel1380's and, to be honest, did understand NOTHING.
\n" ); document.write( "My feeling is that this solution is wrong. Although the answer is correct.\r
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\n" ); document.write( "\n" ); document.write( "Below please find the correct solution.\r
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document.write( "From the Pythagorean theorem we have\r\n" );
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document.write( " = ,\r\n" );
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document.write( "or, applying the formula for square difference,\r\n" );
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document.write( "(c-a)*(c+a) = .\r\n" );
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document.write( "The left side is the product of two positive integer numbers. The right side is a square of a prime number.\r\n" );
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document.write( "It follows from the Fundamental Theorem of Arithmetic (theorem on unique decomposition of any integer into the product of prime numbers)\r\n" );
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document.write( "that one and only one of two alternatives takes place.\r\n" );
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document.write( "1.   EITHER\r\n" );
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document.write( "     c + a = 11,\r\n" );
document.write( "     c - a = 11,\r\n" );
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document.write( "2.   OR\r\n" );
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document.write( "     c + a = 121,\r\n" );
document.write( "     c - a =   1.\r\n" );
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document.write( "The alternative 1 implies a = 0, therefore, it is excluded.\r\n" );
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document.write( "The alternative 2 implies (after adding the two equation) that\r\n" );
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document.write( "    2c = 122  ===>  c  =  = 61.\r\n" );
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document.write( "    Then as the next easy step you find  a = 60.\r\n" );
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\n" ); document.write( "\n" ); document.write( "So, the answer is 61 for the hypotenuse and 60 for the longer leg.\r
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "As a conclusion, I want to say, addressing to the tutors:\r
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document.write( "    The problems like this - they are not routine school math - require very careful treatment,\r\n" );
document.write( "    because they have educational impact to students.\r\n" );
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document.write( "    So,  EITHER  you can solve and present a solution brilliantly, - and then your contribution is positive,\r\n" );
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document.write( "         OR you can not do it in this manner - and then your contribution is negative.\r\n" );
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document.write( "    Notice:  not ZERO contribution, but NEGATIVE, because the student will not obtain that positive charge and charm that the problem \r\n" );
document.write( "    really carries itself.\r\n" );
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document.write( "    Instead he or she  will get the portion of disgusting and the portion of hate to Math.\r\n" );
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