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