What are Pythagorean Triples?
The Pythagorean theorem states that, in a right angle triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem was proposed by a Greek philosopher named Pythagoras. In other words, the square of the side opposite to the right angle is the same as the sum of squares of the remaining two sides.
A pythagorean triple is a set of 3 values corresponding to the 3 sides a, b, and c of a triangle such that a² + b² = c² and they are denoted as (a, b, c). Here, a, b, and c are integers.
It is seen that even (ka, kb ,kc) is a pythagorean triple where k is any positive integer. This is a famous and ancient theory and has been known since Babylonian time.
How can we use Pythagorean triple:
There are 16 primitive (basic) Pythagorean triples for numbers up to 100. Here are some as examples:
(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)
This means any triangle whose sides are any of these pairs, is a right triangle or Pythagorean triangle.
Now, suppose we have a triangle with sides (6, 8,10) or (10, 24, 26) or (15, 36, 39) or (40, 42, 58). These triangles are also right triangles as the corresponding sides are multiples of the primitive triples.
Example: If we consider (3, 4, 5) then 2(3, 4, 5) = (6, 8,10) is a Pythagorean triple
Similarly, if we consider (5, 12, 13) then 3(5, 12, 13) = (15, 36, 39) is also a Pythagorean triple.
We can see that 39² = 15² + 36².
Using the basic rule of Pythagorean triple, we can identify whether the 3 sides form a right triangle or not.
