Final answer:
Using the Pythagorean theorem, the sets 9, 40, 41 and 31, 40, 41 can represent the lengths of the sides of a right triangle since the square of the largest number equals the sum of the squares of the other two numbers.
Step-by-step explanation:
To determine whether a set of three numbers can represent the sides of a right triangle, we utilize the Pythagorean theorem, which states that for any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
The theorem is mathematically expressed as a² + b² = c², where c is the hypotenuse and a and b are the other two sides.
Let's apply this to the provided sets of numbers:
- For the set 12, 15, 21: Checking if 12² + 15² = 21², we get 144 + 225 ≠ 441, hence this set cannot be sides of a right triangle.
- For the set 40, 42, 58: Checking if 40² + 42² = 58², we get 1600 + 1764 ≠ 3364, hence this set also cannot be sides of a right triangle.
- For the set 9, 40, 41: Checking if 9² + 40² = 41², we get 81 + 1600 = 1681, which is indeed 41², so this set can be sides of a right triangle.
- For the set 31, 40, 41: Checking if 31² + 40² = 41², we get 961 + 1600 = 2561, which is indeed 41², so this set can be sides of a right triangle.
The sets 9, 40, 41 and 31, 40, 41 can represent the lengths of the sides of a right triangle.