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.
Sample Math Problems
Problem
Find if (36, 77, 86) is a Pythagorean triple.
Solution
The following calculation should be true:
77² + 36² = 86²
However, we see that 77² + 36² = 1775 and 86² = 7396. Since 1775 ≠ 7396, the given is not a pythagorean triple.
Problem
Is (22, 120, 122) a Pythagorean triple?
Solution
Since all numbers are even we can divide by 2.
Then we see (22,120,122) is a multiple of (11, 60, 61).
Since (11, 60, 61) is a popular Pythagorean triple, (22,120,122) is also a Pythagorean triple. (You can verify (11, 60, 61) is a Pythagorean triple by computing like we did in the previous solved example)
Problem
A student measures the sides of a large triangular shaped field and notes down the side lengths as 56 m, 192 m, and 200 m. Is the field a right triangle?
Solution
We know that (56,192, 200) can be written as 8*(7, 24, 25) - factor out 8.
(7, 24, 25) is a popular Pythagorean triple. (Once again, you can verify this too like we did in the first example)
So, the field is a right triangle.
Problem
If (20, x, 29) is a Pythagorean triple, find x.
Solution
Here 29 is the longest side,
so 29² = 20² + x²
x = square root of (29² - 20²) = 21
