Final answer:
Right triangles with all odd integer lengths do not exist because the sum of the squares of two odd integers is even, making it impossible for the square of the hypotenuse to also be an odd integer, as required by the Pythagorean theorem.
Step-by-step explanation:
The question concerns whether right triangles with all odd integer lengths exist. This is a matter in mathematics related to the Pythagorean theorem, which expresses the relationship between the three sides of a right triangle. The Pythagorean theorem 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), as captured by the equation a² + b² = c².
However, when it comes to a triangle with all sides of odd length, this becomes impossible. The square of an odd number is always odd, and the sum of two odd numbers is always even. Hence, if both a and b are odd (producing odd squares), their sum a² + b² would be even, and cannot equal the square of another odd number (c²). Therefore, the answer is false; right triangles with all sides being odd integers cannot exist because it would violate the Pythagorean theorem.