document.write( "Question 972181: An acute triangle has a perimeter of 18. How many possible triangles are there if all sides have integer values? \n" ); document.write( "
Algebra.Com's Answer #594576 by Edwin McCravy(20056) $\"\"$ $\"About$
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An acute triangle has a perimeter of 18. How many possible triangles are there if all sides have integer values?
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document.write( "There are two things we must make sure of.\r\n" );
document.write( "1. That it is a triangle.\r\n" );
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document.write( "2. That it is an acute triangle.\r\n" );
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document.write( "We can't use \"maximum\" or \"minimum\" because triangles may have 2 or 3 equal\r\n" );
document.write( "sides. So we will use \"minimal\" and \"maximal\".\r\n" );
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document.write( "Minimal side length means the length of a side than is not greater than either\r\n" );
document.write( "of the other two sides.\r\n" );
document.write( "Maximal side length means the length of a side than is not less than either of\r\n" );
document.write( "the other two sides.\r\n" );
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document.write( "1. To insure that it is a triangle:\r\n" );
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document.write( "The minimal side length must be greater than the difference between the two maximal side lengths.\r\n" );
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document.write( "2. To insure that it is an acute triangle:\r\n" );
document.write( "The maximal length of a side is less than the hypotenuse of a right triangle\r\n" );
document.write( "with the two minimal sides as legs.\r\n" );
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document.write( "Case 1: Minimal side is 1.\r\n" );
document.write( "If the minimal side were 1, the two maximal sides would have sum 17.\r\n" );
document.write( "Their difference would have to be smaller than 1.  But the closest together the\r\n" );
document.write( "other sides could be would be when they are 9 and 8, but that difference is 1.\r\n" );
document.write( "So the smallest side cannot be 1.\r\n" );
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document.write( "Case 2: Minimal side is 2.\r\n" );
document.write( "If the minimal side were 2, the two sides would have sum 16.\r\n" );
document.write( "They can be 8 and 8 because 2 is greater than their difference which is 0.\r\n" );
document.write( "But they cannot be any farther apart because 9 and 7 have difference 2.\r\n" );
document.write( "We check to see if the maximal length of a side is less than the\r\n" );
document.write( "hypotenuse of a right triangle with the two minimal sides as legs.\r\n" );
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document.write( "The maximal side 8 is less than 8.2.\r\n" );
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document.write( "So sides of 2,8,8 is the only solution with minimal side 2.\r\n" );
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document.write( "Case 3: Minimal side is 3\r\n" );
document.write( "If the minimal side were 3, the two maximal sides would have sum 15.\r\n" );
document.write( "They can be 8 and 7 because 3 is greater than their difference which is 2.\r\n" );
document.write( "But they cannot be any farther apart because 9 and 6 have difference 3.\r\n" );
document.write( "We check to see if the maximal length of a side is less than the\r\n" );
document.write( "hypotenuse of a right triangle with the two minimal sides as legs.\r\n" );
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document.write( "But the maximal side 8 is greater than 7.6. \r\n" );
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document.write( "So sides of 3,7,8 is not a solution because it is an obtuse triangle.\r\n" );
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document.write( "Case 4: Minimal side is 4\r\n" );
document.write( "If the minimal side were 4, the two maximal sides would have sum 14.\r\n" );
document.write( "They can be 7 and 7 because 4 is greater than their difference which is 0.\r\n" );
document.write( "Tney can also be 8 and 6 because 4 is greater than their difference which is 2.\r\n" );
document.write( "But they cannot be any farther apart because 9 and 5 have difference 4.\r\n" );
document.write( "We check each of these to see if the maximal length of a side is less than the\r\n" );
document.write( "hypotenuse of a right triangle with the two minimal sides as legs.\r\n" );
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document.write( "The maximal side 7 is less than 8.1. \r\n" );
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document.write( "So sides of 4,7,7 is a solution\r\n" );
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document.write( "The maximal side 8 is greater than 7.2. \r\n" );
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document.write( "So sides of 4,6,8 is the only solution with minimal side 4.\r\n" );
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document.write( "Case 5: Minimal side is 5\r\n" );
document.write( "If the minimal side were 5, the two maximal sides would have sum 13.\r\n" );
document.write( "They can be 7 and 6 because 5 is greater than their difference which is 1.\r\n" );
document.write( "Tney can also be 8 and 5 because 5 is greater than their difference which is 3.\r\n" );
document.write( "But they cannot be any farther apart because 9 and 4 have difference 5.\r\n" );
document.write( "We check each of these to see if the maximal length of a side is less than the\r\n" );
document.write( "hypotenuse of a right triangle with the two minimal sides as legs.\r\n" );
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document.write( "The maximal side 7 is less than 7.4. \r\n" );
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document.write( "So sides of 5,6,7 is a solution.\r\n" );
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document.write( "The maximal side 8 is greater than 7.1.\r\n" );
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document.write( "So sides of 5,5,8 is not a solution, for it is an obtuse triangle.\r\n" );
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document.write( "Case 6. Minimal side is 6\r\n" );
document.write( "If the minimal side were 6, the two maximal sides would have sum 12.\r\n" );
document.write( "They can be 6 and 6 because 6 is greater than their difference which is 0.\r\n" );
document.write( "They can only be 6 and 6 because if one side were greater than 6 the other \r\n" );
document.write( "side would smaller than 6 and 6 wouldn't be the minimal side.\r\n" );
document.write( "We know that 6,6,6, is a solution because it's equilateral, and an\r\n" );
document.write( "equilateral triangle is acute, since all three angles are 60°.\r\n" );
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document.write( "So there are 4 solutions:\r\n" );
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document.write( "1. Sides 2,8,8\r\n" );
document.write( "2. Sides 4,7,7\r\n" );
document.write( "3. Sides 5,6,7\r\n" );
document.write( "4. Sides 6,6,6\r\n" );
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document.write( "Edwin
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