document.write( "Question 118665: How many non-congruent triangles, each with a perimeter of 15 units, can be constructed with all integral side lengths? \n" ); document.write( "

Algebra.Com's Answer #86884 by Edwin McCravy(20056) $\"\"$ $\"About$

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How many non-congruent triangles, each with a perimeter of 15 units, can be constructed with all integral side lengths?\r
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document.write( "We have to find all choices of positive \r\n" );
document.write( "integers x, y, and z such that\r\n" );
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document.write( "x + y + z = 15\r\n" );
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document.write( "x + y + z = 15\r\n" );
document.write( "x + y > z\r\n" );
document.write( "x + z > y\r\n" );
document.write( "y + z > x\r\n" );
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document.write( "From\r\n" );
document.write( "x + y + z = 15\r\n" );
document.write( "    x + y = 15 - z\r\n" );
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document.write( "and since x + y > z,\r\n" );
document.write( "         15 - z > z\r\n" );
document.write( "             15 > 2z\r\n" );
document.write( "            7.5 > z\r\n" );
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document.write( "From\r\n" );
document.write( "x + y + z = 15\r\n" );
document.write( "    x + z = 15 - y\r\n" );
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document.write( "and since x + z > y,\r\n" );
document.write( "         15 - y > y\r\n" );
document.write( "             15 > 2y\r\n" );
document.write( "            7.5 > y\r\n" );
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document.write( "From\r\n" );
document.write( "x + y + z = 15\r\n" );
document.write( "    y + z = 15 - x\r\n" );
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document.write( "and since y + z > x,\r\n" );
document.write( "         15 - x > x\r\n" );
document.write( "             15 > 2x\r\n" );
document.write( "            7.5 > x\r\n" );
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document.write( "So all the sides are 7 or less\r\n" );
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document.write( "And since to make sure they are non-congruent, \r\n" );
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document.write( "we'll require that x < y < z, so\r\n" );
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document.write( "x = 1 y = 7 z = 7\r\n" );
document.write( "x = 2 y = 6 z = 7\r\n" );
document.write( "x = 3 y = 5 z = 7\r\n" );
document.write( "x = 3 y = 6 z = 6\r\n" );
document.write( "x = 4 y = 4 z = 7\r\n" );
document.write( "x = 4 y = 5 z = 6\r\n" );
document.write( "x = 5 y = 5 z = 5\r\n" );
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document.write( "So there are 7.\r\n" );
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document.write( "Edwin

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