The sets of numbers 6, 8, 10 and 5, 12, 13 are Pythagorean triples. Use what you know about the Pythagorean Theorem and explain or show why they are Pythagorean triples. Be sure to show your work for each set of triples! (5 points)
To determine if a set of numbers forms a Pythagorean triple, we need to apply the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
For the first set of numbers 6, 8, and 10, we need to check if:
6^2 + 8^2 = 10^2
36 + 64 = 100
100 = 100
Since both sides of the equation are equal, the first set of numbers forms a Pythagorean triple.
For the second set of numbers 5, 12, and 13, we need to check if:
5^2 + 12^2 = 13^2
25 + 144 = 169
169 = 169
Again, both sides of the equation are equal, so the second set of numbers forms a Pythagorean triple as well.
In summary, both sets of numbers satisfy the Pythagorean Theorem, which is why they are considered Pythagorean triples.