document.write( "Question 538701: How many different isosceles triangles have integer side lengths and a perimeter of 81 units? \n" ); document.write( "
Algebra.Com's Answer #353237 by AnlytcPhil(1806) $\"\"$ $\"About$
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document.write( "Let's determine first what the largest possible measure each of \r\n" );
document.write( "the two congruent legs can have:\r\n" );
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document.write( "The smallest the base can be is 1, so the two congruent legs\r\n" );
document.write( "would have length  or 40 units each\r\n" );
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document.write( "Now we will determine what the smallest possible measure each of \r\n" );
document.write( "the two congruent legs can have:\r\n" );
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document.write( "If the two legs are \"a\" units each and the base is c, then by the\r\n" );
document.write( "triangle inequality:\r\n" );
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document.write( "(1)                2a > c\r\n" );
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document.write( "and since the perimeter = 81 = a + a + c, \r\n" );
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document.write( "               2a + c = 81\r\n" );
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document.write( "(2)                 c = 81 - 2a\r\n" );
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document.write( "Substitute the 81 - 2a for c in equation (1)\r\n" );
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document.write( "                   2a > c\r\n" );
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document.write( "                   2a > 81 - 2a\r\n" );
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document.write( "                   4a > 81\r\n" );
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document.write( "                    a > \r\n" );
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document.write( "                    a > \r\n" );
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document.write( "Therefore the smallest \"a\" can be is 21\r\n" );
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document.write( "So \"a\" can range from 21 to 40, inclusive.\r\n" );
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document.write( "There are 40 integers from 1 to 40, and 20 of those are smaller than 21.\r\n" );
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document.write( "So there are 40-20 or 20 possible triangles.  Here they all are:\r\n" );
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document.write( "                  \r\n" );
document.write( "1. sides 21, 21, 39\r\n" );
document.write( "2. sides 22, 22, 37\r\n" );
document.write( "3. sides 23, 23, 35\r\n" );
document.write( "4. sides 24, 24, 33\r\n" );
document.write( "5. sides 25, 25, 31\r\n" );
document.write( "6. sides 26, 26, 29\r\n" );
document.write( "7. sides 27, 27, 27\r\n" );
document.write( "8. sides 28, 28, 25\r\n" );
document.write( "9. sides 29, 29, 23\r\n" );
document.write( "10. sides 30, 30, 21\r\n" );
document.write( "11. sides 31, 31, 19\r\n" );
document.write( "12. sides 32, 32, 17\r\n" );
document.write( "13. sides 33, 33, 15\r\n" );
document.write( "14. sides 34, 34, 13\r\n" );
document.write( "15. sides 35, 35, 11\r\n" );
document.write( "16. sides 36, 36, 9\r\n" );
document.write( "17. sides 37, 37, 7\r\n" );
document.write( "18. sides 38, 38, 5\r\n" );
document.write( "19. sides 39, 39, 3\r\n" );
document.write( "20. sides 40, 40, 1\r\n" );
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document.write( "Edwin
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