SOLUTION: can you help here . : how many triangles can there be if two sides have 15 cm and 12cm and all sides have integral lengths ?

Question 530903: can you help here . : how many triangles can there be if two sides have 15 cm and 12cm and all sides have integral lengths ?

Answer by oberobic(2304) (Show Source): You can put this solution on YOUR website!
The third side cannot be >= the sum of the other two sides or less than their difference.
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The sides are 15 and 12.
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If the third side is 3, then the other two sides would have to be lying one on top of the other for the lines to touch. So that would not be a triangle. If the third side was ever so slightly more than 3, say 3.0001, then a triangle could be constructed. Of course, it would be have a very small angle between the sides that are 15 and 12.
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Similarly, if the two sides were placed as wide apart as possible (say, 179.9999 degrees), the third side would be less than 27. If the third side was 27, then the other two sides would have to be 180 degrees apart, which means the lines would be lying end to end and it would not be triangle.
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So, 3 < x < 27, where x = number of integral sides
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x can be 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, and 26.
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Therefore, there can be 23 triangles that have sides of 15, 12, and one of these integral values of x.
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Done.
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