Pythagorean Triples consist of sets of three positive integers a, b, and c, which satisfy the equation a2 + b2 = c2. In these triples, c represents the hypotenuse (the longest side) of a right triangle, while a and b are the other two sides. This relationship stems from the Pythagorean Theorem, where the square of the longest side equals the sum of the squares of the other two sides.
Table
| (3, 4, 5) | (5, 12, 13) | (8, 15, 17) | (7, 24, 25) |
| (20, 21, 29) | (12, 35, 37) | (9, 40, 41) | (28, 45, 53) |
| (11, 60, 61) | (16, 63, 65) | (33, 56, 65) | (48, 55, 73) |
| (13, 84, 85) | (36, 77, 85) | (39, 80, 89) | (65, 72, 97) |
Also Check: Types of angles
Pythagorean Triples Formula
The formula for generating Pythagorean triples is:
If \( m \) and \( n \) are positive integers such that \( m > n > 0 \), then:
- a = m^2 - n^2
- b = 2mn
- c = m^2 + n^2
Here,
- a, b, and c are the sides of the Pythagorean triple.
- The integers \( m \) and \( n \) must be chosen such that they are coprime (no common factors) and \( m \) is greater than \( n \).
- This formula ensures that \( a^2 + b^2 = c^2 \), satisfying the Pythagorean theorem.
For example, choosing \( m = 2 \) and \( n = 1 \):
- a = 2^2 - 1^2 = 3
- b = 2 \times 2 \times 1 = 4
- c = 2^2 + 1^2 = 5
Thus, the Pythagorean triple formed is (3, 4, 5).
Pythagorean triples are fundamental in number theory and have applications in various fields including geometry and cryptography.
Also Check: Continuous Variable
How Do Pythagorean Triples Form?
The number can be an odd or even number, as we all know. Let's look at how to make
Pythagorean triples now.
Case 1: When the number is odd
Consider an odd number, denoted as x.
For an odd x, the Pythagorean triple can be expressed as x, (x2/2) - 0.5, (x2/2) + 0.5.
Let's illustrate this with an example: (7, 24, 25).
Here, x = 7, which is an odd number.
Calculating the Pythagorean triple:
- (x2/2) - 0.5 = (49/2) - 0.5 = 24.5 - 0.5 = 24
- (x2/2) + 0.5 = (49/2) + 0.5 = 24.5 + 0.5 = 25
Thus, the Pythagorean triple formed is (7, 24, 25).
Case 2: Even Number
If "x" is an even number, the corresponding Pythagorean triple is formed as follows: (x, x/22 - 1, x/22 + 1).
For instance, consider the example (16, 63, 65).
To construct this Pythagorean triple:
- Given that x = 16 (an even number),
- x/22 - 1 = 16/22 - 1 = 82 - 1 = 64 - 1 = 63
- x/22 + 1 = 16/22 + 1 = 82 + 1 = 64 + 1 = 65
Thus, the Pythagorean triple formed is (16, 63, 65).
Frequently Asked Questions
If the numbers A, B, and C form a Pythagorean triplet, it means that A^2 + B^2 = C^2. This means the three numbers represent the side lengths of a right triangle, where A and B are the shorter sides and C is the longer, hypotenuse side.
There is no single largest Pythagorean triplet, as Pythagorean triples can be scaled up indefinitely. However, the largest known primitive Pythagorean triplet (where the three numbers have no common factors) is (99,990, 100,010, 141,420).
The formula for generating Pythagorean triples is: (a, b, c) = (m^2 - n^2, 2mn, m^2 + n^2), where m and n are positive integers and m > n. This formula ensures that a^2 + b^2 = c^2, satisfying the Pythagorean theorem.
Yes, 9, 12, and 15 form a Pythagorean triplet, as 9^2 + 12^2 = 81 + 144 = 225, which is equal to 15^2.
Yes, 15, 20, and 25 form a Pythagorean triple, as 15^2 + 20^2 = 225 + 400 = 625, which is equal to 25^2.