The sets of numbers are 6,8,10 and 5,12,13 are Pythagorean triples. Use what you know about the Pythagoream theorem snd explain or show why they are Pythagorean triples
To show that a set of numbers is a Pythagorean triple, we need to verify that it satisfies the Pythagorean theorem, which states that in a 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 other two sides.
Let's start with the set of numbers 6, 8, and 10.
If we assume that 10 is the hypotenuse, we can calculate the squares of the other two sides:
6^2 = 36
8^2 = 64
The sum of the squares of these two sides is 36 + 64 = 100.
Now, let's calculate the square of the hypotenuse (10^2):
10^2 = 100
As we can see, the sum of the squares of the other two sides (36 + 64 = 100) is indeed equal to the square of the hypotenuse (100). Thus, the set of numbers 6, 8, and 10 satisfies the Pythagorean theorem and is a Pythagorean triple.
Now, let's move on to the set of numbers 5, 12, and 13.
If we assume that 13 is the hypotenuse, we can calculate the squares of the other two sides:
5^2 = 25
12^2 = 144
The sum of the squares of these two sides is 25 + 144 = 169.
Now, let's calculate the square of the hypotenuse (13^2):
13^2 = 169
Again, we can see that the sum of the squares of the other two sides (25 + 144 = 169) is equal to the square of the hypotenuse (169). Therefore, the set of numbers 5, 12, and 13 also satisfies the Pythagorean theorem and is a Pythagorean triple.