I want to know the number of triangles
Geometry
Level
4
Find the number of non-degenerate triangles such that at least one of its side length is 15, and the other two sides have an integer length of 15 or less.
Note: Count only non-congruent triangles.
The answer is 64.
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1 solution
We want to find the number of distinct triangles of sides ( a , b , 1 5 ) , subject to a ≤ b ≤ 1 5 .
The triangle inequality states that a + b > 1 5 .
We can check that:
If b ≤ 7 , then there are no solutions.
If b = 8 , then there is 1 solution.
If b = 9 , then there are 3 solutions.
If b = 1 0 , then there are 5 solutions.
If b = 1 1 , then there are 7 solutions.
If b = 1 2 , then there are 9 solutions.
If b = 1 3 , then there are 11 solutions.
If b = 1 4 , then there are 13 solutions.
If b = 1 5 , then there are 15 solutions.
Hence, in total, there are 64 solutions.