document.write( "Question 1182225: Can the converse of the Pythagorean theorem determine which triangle with the given three side lengths is a right triangle.\r
\n" ); document.write( "\n" ); document.write( "16,21,24\r
\n" ); document.write( "\n" ); document.write( "4,9,12\r
\n" ); document.write( "\n" ); document.write( "20,21,29\r
\n" ); document.write( "\n" ); document.write( "5,12,14 \n" ); document.write( "

Algebra.Com's Answer #853566 by ikleyn(53615) $\"\"$ $\"About$

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\n" ); document.write( "Can the converse of the Pythagorean theorem determine which triangle
\n" ); document.write( "with the given three side lengths is a right triangle.
\n" ); document.write( "(a) 16,21,24
\n" ); document.write( "(b) 4,9,12
\n" ); document.write( "(c) 20,21,29
\n" ); document.write( "(d) 5,12,14
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\n" ); document.write( "\n" ); document.write( "Yes, it can.\r
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\n" ); document.write( "\n" ); document.write( "Regarding this concrete problem, the most part of options (a) - (d) can be analyzed MENTALLY,
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document.write( "(a)  We want to check if \r\n" );
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document.write( "         16^2 + 21^2 = 24^2.\r\n" );
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document.write( "     In this hypothetical equality, two terms, 16^2 and 24^2, are even integer numbers, while 21^2 \r\n" );
document.write( "     is odd integer number.  Hence, this equality is not possible - this triangle is not a right triangle.\r\n" );
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document.write( "(b)  We want to check if \r\n" );
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document.write( "         4^2 + 9^2 = 12^2.\r\n" );
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document.write( "     In this hypothetical equality, two terms, 4^2 and 12^2, are even integer numbers, while 9^2 \r\n" );
document.write( "     is odd integer number.  Hence, this equality is not possible - this triangle is not a right triangle.\r\n" );
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document.write( "(d)  We want to check if \r\n" );
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document.write( "         5^2 + 12^2 = 14^2.\r\n" );
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document.write( "     In this hypothetical equality, two terms, 12^2 and 14^2, are even integer numbers, while 5^2 \r\n" );
document.write( "     is odd integer number.  Hence, this equality is not possible - this triangle is not a right triangle.\r\n" );
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document.write( "(c)  We want to check if \r\n" );
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document.write( "         20^2 + 21^2 = 29^2.\r\n" );
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document.write( "     In this case, this reasoning with odd-even numbers does not work, so we should check it performing \r\n" );
document.write( "     direct explicit computation.\r\n" );
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document.write( "     Left side is  20^2 + 21^2 = 400+ 441 = 841.  Right side is 29^2 = (30-1)^2 = 900 - 2*30 + 1 = 841.\r\n" );
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document.write( "     Both sides are equal - hence, this triangle is a right-angled triangle.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved MENTALLY, from the beginning to the end !\r
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