Consecutive side lengths
Geometry
Level
2
It is possible to construct a right triangle with consecutive integers side lengths, namely, 3-4-5.
Is there more than 1 way to construct a right triangle with consecutive integers side lengths?
No
Yes
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1 solution
Apply the Pythagorean relationship to the second triangle:
( n 2 ) + ( n 2 + 2 n + 1 ) = n 2 + 4 n + 4 n 2 − 2 n − 3 = 0 ( n − 3 ) ( n + 1 ) = 0
This gives ( n = 3 ) and ( n = − 1 ) as solutions. Obviously, the negative solution is inadmissible, leaving the 3-4-5 triangle as the only possible right triangle with consecutive integer side lengths.