Question 1204770: How many scalene triangles have perimeter less than 16 and sides of integral length?
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Reaching the correct answer to a problem like this requires making an organized list, so that no triangle is overlooked and no triangle is listed twice.
The perimeter has to be less than 16, so start with 15 and work down.
15: If the perimeter is 15, the longest any side can be is 7. Look for solutions with longest side 7. The other two side lengths must be different integers whose sum is 8.
First solutions: (7,6,2) and (7,5,3)
Next look for other triangles with perimeter 15 and longest side 6. The other two side lengths must be different integers with a sum of 9.
Next solution: (6,5,4)
14: If the perimeter is 14, the longest any side can be is 6. Look for solutions with longest side 6. The other two side lengths must be different integers whose sum is 8.
Next solution: (6,5,3)
13: If the perimeter is 13, the longest any side can be is again 6. Look for solutions with longest side 6. The other two side lengths must be different integers whose sum is 7.
Next solutions: (6,5,2) and (6,4,3)
12: If the perimeter is 12, the longest any side can be is 5. Look for solutions with longest side 5. The other two side lengths must be different integers whose sum is 7.
Next solution: (5,4,3)
11: If the perimeter is 11, the longest any side can be is again 5. Look for solutions with longest side 5. The other two side lengths must be different integers whose sum is 6.
Next solution: (5,4,2)
10: If the perimeter is 10, the longest any side can be is 4. Look for solutions with longest side 4. The other two side lengths must be different integers whose sum is 6.
No solutions here; neither (4,4,2) nor (4,3,3) is scalene.
9: If the perimeter is 9, the longest any side can be is again 4. Look for solutions with longest side 4. The other two side lengths must be different integers whose sum is 5.
Next solution: (4,3,2)
No scalene triangle with integer side lengths has a side of length 1, so there are no more solutions.
We found....
(1) (7,6,2)
(2) (7,5,3)
(3) (6,5,4)
(4) (6,5,3)
(5) (6,5,2)
(6) (6,4,3)
(7) (5,4,3)
(8) (5,4,2)
(9) (4,3,2)
ANSWER: 9
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