Exploring Pythagorean Triples | Sets That Satisfy the Pythagorean Theorem and Their Significance in Geometry and Trigonometry

{3,4,5}{5,12,13}{8,15,17}{7,24,25}

The sets you provided are known as Pythagorean triples

The sets you provided are known as Pythagorean triples. A Pythagorean triple consists of three positive integers (a, b, c) that satisfy 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 analyze each set you provided:

1. Set {3, 4, 5}
In this set, the three numbers are 3, 4, and 5. These numbers form a Pythagorean triple because 3^2 + 4^2 = 5^2. So, if you have a right triangle with side lengths of 3 and 4, the length of the hypotenuse will be 5.

2. Set {5, 12, 13}
Similarly, the set {5, 12, 13} forms a Pythagorean triple since 5^2 + 12^2 = 13^2. Hence, a right triangle with side lengths 5 and 12 will have a hypotenuse of length 13.

3. Set {8, 15, 17}
Again, this set {8, 15, 17} is a Pythagorean triple as 8^2 + 15^2 = 17^2. Hence, a right triangle with side lengths 8 and 15 will have a hypotenuse of length 17.

4. Set {7, 24, 25}
Lastly, the set {7, 24, 25} is also a Pythagorean triple as 7^2 + 24^2 = 25^2. Therefore, if you have a right triangle with side lengths 7 and 24, the length of the hypotenuse will be 25.

These Pythagorean triples have been studied extensively since ancient times, and they have numerous applications in geometry, trigonometry, and even cryptography.

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