To show that a set of numbers is a Pythagorean triple, we need to verify that the Pythagorean theorem holds true for that set.
For the set of numbers 7, 24, and 25:
According to the Pythagorean theorem, for any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, 7 and 24 are the two sides, and 25 is the hypotenuse.
Let's square each of the numbers:
7^2 = 49
24^2 = 576
25^2 = 625
Now, let's check if the Pythagorean theorem holds true:
49 + 576 = 625
Since the sum of the squares of the two sides (49 + 576) equals the square of the hypotenuse (625), the set of numbers (7, 24, 25) is a Pythagorean triple.
For the set of numbers 9, 40, and 41:
Using the same process, let's square each of the numbers:
9^2 = 81
40^2 = 1600
41^2 = 1681
Now, let's check if the Pythagorean theorem holds true:
81 + 1600 = 1681
Since the sum of the squares of the two sides (81 + 1600) equals the square of the hypotenuse (1681), the set of numbers (9, 40, 41) is also a Pythagorean triple.