Pythagorean Triples

Pythagorean triples are an important concept in two-dimensional geometry that plays a crucial part in number theory and geometry. Pythagorean triples are sets of non-negative integers (a, b, c) that satisfy the Pythagorean theorem ( a² + b² = c² ). Here, “c” represents the hypotenuse of a right-angled triangle. In this detailed guide, we will go through the Pythagorean triples definition, methods for generating Pythagorean triples, their formulas, and various examples.

1.0What Are Pythagorean Triples?

The Pythagorean triples definition stands for three non-negative integers that completely satisfy the Pythagorean theorem for a right triangle. Mathematically speaking, the Pythagorean triple is a set of three positive integers (a, b, c) that satisfy the equation a² + b² = c². 

In this equation, “a” and “b” are the legs of the triangle, whereas “c” is the hypotenuse. 

2.0Pythagorean Triangles

Triangles that have sides forming a Pythagorean triple are called Pythagorean triangles. Pythagoras's theorem states that “In any right-angled triangle, the square of the hypotenuse (biggest side) is always equal to the sum of squares of the other two adjacent sides.” Triangles that satisfy this theorem can be considered Pythagorean triangles 

3.0Common List of Pythagorean Triples

The most well-known list of Pythagorean triples includes values like:

(3, 4, 5), (5, 12, 13), (13, 84, 85), (7, 24, 25), (8, 15, 17), (9, 40, 41)

These sets satisfy the Pythagorean theorem, as shown below:

3² + 4² = 5²   ( 9 + 16 = 25)

And 

5² + 12² = 13²  (25 + 144 = 169)

4.0Pythagorean Triples Formula

The Pythagorean triples formula is used to find triples that satisfy the Pythagoras theorem and the sides of right-angled triangles. We know that when a, b, & c are the perpendicular, base, & hypotenuse of a right-angled triangle, then by Pythagoras' theorem, we have: c² = a² + b². 

Therefore, the Pythagorean triples formula can be given as,

• P = m² – n²
• B = 2mn
• H = m² + n²

where,

• P, B = Perpendicular & Base of a right-angled triangle
• H = Hypotenuse of a right-angled triangle
• m & n are any two positive integers; where m > n
• m & n are coprime, & both should not be odd numbers

Let’s look at some mathematical examples of the Pythagorean triples formula to understand this better.

If m = 3 and n = 2

Then

( 3² - 2², 2 x 3 x 2, 3² + 2² ) = ( 9 - 4, 12, 9 + 4) 

= ( 5, 12, 13 ) 

This confirms that (5, 12, 13) is a Pythagorean triple.

5.0Generating Pythagorean Triples

There are many ways of generating Pythagorean triples other than using the common formula. 

• Scaling existing triples: For generating Pythagorean triples by scaling existing ones, multiply all elements by a common integer factor k to get another valid Pythagorean triple. If (a, b, c) is a Pythagorean triple, then (ka, kb, kc) is also a Pythagorean triple for any positive integer k. 
• Finding primitive triples: Use the coprime values for m and n for generating Pythagorean triples. 

6.0Pythagorean Triples List from 1 to 100

The following table of Pythagorean triples list from 1 to 100 has the Pythagorean triples of integers where the values are less than 100. 

7.0Examples of Pythagorean Triples

Let us look at some examples of Pythagorean triples to see how you can utilise what you learnt in mathematical problems. 

Example 1: Prove that (5, 12, 13) is a Pythagorean triple.

Solution:

To prove: (5, 12, 13) is a Pythagorean Triple

We know that, a² + b²  = c²

(a, b, c) = (5, 12, 13)

Now, substitute the values, 

5² + 12² = 13²

25 + 144 = 169

169 = 169

Hence, the given integer set satisfies the Pythagoras theorem; (5, 12, 13) is Pythagorean triples.

Example 2: Check with the help of the Pythagorean theorem if 7, 15, & 17 are Pythagorean triplets.

Solution:

(Perpendicular, Base, Hypotenuse) = (7, 15, 17) (Hypotenuse is always the biggest side/integer)

We know that P² + B²  = H²

By substituting the values in the equation, we get

7² + 15² = 17²

49 + 225 =  289

274 ≠ 289 (LHS not equal to RHS)

Hence, the given integers do not satisfy the Pythagoras theorem. Therefore, "7, 15, 17" is not a Pythagorean triplet.

8.0Conclusion

Pythagorean triples play a significant role in mathematics, especially in geometry and number theory. The concepts discussed in the guide would serve as a great reference for solving problems related to the right triangle.