Final answer:
The hypotenuse of a right triangle with legs 1 and a positive integer n is irrational. This follows from the Pythagorean theorem, which results in an irrational number when taking the square root of a sum that is not a perfect square.
Step-by-step explanation:
If n is a positive integer, then the hypotenuse of a right triangle with legs 1 and n is indeed irrational. We can verify this statement using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The formula is given by a² + b² = c². For our triangle with legs of length 1 and n, this becomes 1² + n² = c², which simplifies to 1 + n² = c². Solving for c, we find that c = √(1 + n²).
Since n is an integer, n² is also an integer. Therefore, 1 + n² is an integer, and unless 1 + n² is a perfect square (which it cannot be for any positive integer n), the square root of 1 + n² will be irrational. This is because the square root of a non-perfect square integer is always an irrational number.