Rare Triangles
These three lines are random. Their individual lengths can range from 1 to 25 integer units. {1,2,3,...,24,25}
For example: Red=19, Green=15 and Blue=3, is one from 25^3 = 15625 permutations.
If we connect 3 lines we can form a triangle, but notice that (19,15,3) would not form a triangle, because the sum of 2 sides must be bigger than the third, and 15 + 3 < 19.
If you list all possibilities, how many are right-angled triangles ?
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1 solution
There are only four Primitive Pythagorean Triples with all numbers (side lengths) less than or equal to 25. Namely,
3 − 4 − 5 5 − 1 2 − 1 3 7 − 2 4 − 2 5 8 − 1 5 − 1 7 . From the first triple (3-4-5), we can have four more triples taking multiples: 6 − 8 − 1 0 9 − 1 2 − 1 5 1 2 − 1 6 − 2 0 1 5 − 2 0 − 2 5 , thus, summing up to a number of 8 triples.
Considering that there are 6 permutation of each one of all these triples to create differently coloured triangles, we come to a total of 6 × 8 = 4 8 right-angled triangles.