Pythagorean Triples are a set of 3 positive integers, namely a, b, and c that perfectly satisfy the Pythagorean Theorem rule:
a2 + b2 = c2, here a, b, and c are the 3 sides of a right angle triangle
In other way, we can say when the 3 sides of a triangle are a Pythagorean Triple; it is a right angle triangle. Only positive integers are found to form Pythagorean Triples.
The smallest Pythagorean Triple is the set (3, 4, 5). Thus, the length of the sides of the triangle is exactly 3, 4, and 5 that satisfies the Pythagorean Theorem rule a2 + b2 = c2
A quick way to find more Pythagorean triples is to multiply all the original terms with the same positive integer. If we continue to scale up the Pythagorean Triples, we will obtain a list of Pythagorean Triples.
For example,
If we multiply the smallestPythagorean Triple (3, 4, 5) with 2, we get,
(3 × 2), (4 × 2), (5 × 2),
=> (6, 8, 10) as the next Pythagorean Triple
Similarly, if we multiply Pythagorean Triple (3, 4, 5) with 3, we get another triple as (3, 12,15). In this way we can obtain a list of Pythagorean Triple by scaling up the smallest triple.
There are 2 types of Pythagorean Triples:
Primitive Pythagorean Triples
Non-primitive Pythagorean Triples
Primitive Pythagorean Triples
A primitive Pythagorean Triple, also known as reduced triple, is a set of positive integers (a, b, c) with a greatest common factor (GCF) of 1. Primitive Pythagorean Triple will always have 1 even number and the value of c will always be odd. Many primitive Pythagorean Triples have 2 prime numbers.
Examples
(3, 4, 5) Triple
GCF = 1
Here, a = 3, b = 4, c = 5
Now,
a2 + b2
=> 32 + 42
=> 9 + 16
=> 25
Again,
c2 => 52 => 25
Thus, the smallestPythagorean Triple is primitive since (3, 4, 5) is a primitive Pythagorean Triple. Some more examples of primitive Pythagorean Triples are (5, 12, 13), (15, 8, 17), (7, 24, 25), (9, 40, 41), (21, 20, 29), and (13, 84, 84).
Non- primitive Pythagorean Triples
A non-primitive Pythagorean Triple, also known as imperative Pythagorean Triple is a set of positive integers (a, b, c) having a GCF larger than 1.
Examples
(6, 8, 10)
GCF = 2
Here, a = 6, b = 8, c = 10
Now,
a2 + b2
=> 62 + 82
=> 36 + 64
=> 100
Again,
c2 => 102 => 100
Some more examples of primitive Pythagorean Triples are (6,8,10), (16, 30, 34), (32,60,68),
Some common Pythagorean Triples not including scaled-up versions is given below:
(3, 4, 5)
(5, 12, 13)
(7, 24, 25)
(8, 15, 17)
(9, 40, 41)
(11, 60, 61)
(12, 35, 37)
(13, 84, 85)
(16, 63, 65)
(20, 21, 29)
(28, 45, 53)
(33, 56, 65)
(36, 77, 85)
(39, 80, 89)
(48, 55, 73)
(65, 72, 97)
Formula: Generating Pythagorean Triples
We can find Pythagorean Triples by using a formula known as Pythagorean triple checker. The formula that can generate both primitive and non-primitive Pythagorean Triples is given below:
Pythagorean Triples Formula
Let us find out if it is possible to find a Pythagorean Triple with integers 1 and 2 as both are positive integers and one is greater than the other
Let, m = 2 and n = 1
Since, we know the values of ‘m’ and ‘n’; we will substitute the values in the above formula to find the side lengths of the right triangle
Solving for a
As we know,
a = m2 – n2
= 22 – 12
= 4 -1
= 3
Solving for b
As we know,
b = 2mn
= 2 × 2 × 1
= 4
Solving for c
As we know,
c = m2 + n2
= 22 + 12
= 4 +1
= 5
As we know, (3, 4, 5) is the smallest Pythagorean Triple, and thus satisfy the Pythagorean Theorem rule
a2 + b2 = c2
Solved Examples
E.g 1)
Is (6, 8, 10) a Pythagorean Triple?
Solution:
Let a = 6, b = 8, c = 10 As we know, to be a Pythagorean triple, it must satisfy the Pythagorean Theorem rule: a2 + b2 = c2 => 62 + 82 = 102 => 36 + 64 = 100 => 100 = 100 Thus, (6, 8, 10) a Pythagorean Triple
Find if (9 12 15) is a Pythagorean Triple.
Solution:
Let a = 9, b = 12, c = 15 As we know, to be a Pythagorean triple, it must satisfy the Pythagorean Theorem rule: a2 + b2 = c2 => 92 + 122 = 152 => 81 + 144 = 225 => 225 = 225 Thus, (9, 12, 15) is a Pythagorean Triple
Find the Pythagorean triple if the hypotenuse of the right triangle is 25 cm.
Solution:
As we know, (a, b, c) = [(m2 – n2), (2mn), (m2 + n2)], here c = 25 cm Thus, m2 + n2 = 25
Generate a Pythagorean triple from two positive integers 5 and 3.
Solution:
From the given question, both are positive integers and one is greater than the other Let, m = 5 and n = 3 Since, we know the values of ‘m’ and ‘n’; we will substitute the values in the above formula to find the side lengths of the right triangle Solving for a As we know, a = m2 – n2 = 52 – 32 = 25 -9 = 16 Solving for b As we know, b = 2mn = 2 × 5 × 3 = 30 Solving for c As we know, c = m2 + n2 = 52 + 32 = 25 +9 = 34 As we know, to be a Pythagorean triple, it must satisfy the Pythagorean Theorem rule: a2 + b2 = c2 => 162 + 302 = 342 => 256 + 900 = 1156 => 1156 = 1156 Thus, the Pythagorean triple obtained from two positive integers 5 and 3 is (16, 30, 34)
Find the possible value of ‘a’ in the Pythagorean Triple (a, 35, 37).
Solution:
Since (a, 35, 37) is a Pythagorean Triple, it must satisfy the Pythagorean Theorem rule: a2 + b2 = c2, here b = 35, c = 37 => a2 + 352 = 372 => a2 = 372 – 352 => a2 = 144 => a = 12 Thus, the value of ‘a’ in the Pythagorean Triple (a, 35, 37) is 12
Applications
Pythagorean Triples are whole numbers that make the Pythagorean Theorem true. It thus helps to find the unknown side of a right triangle when the length of the other 2 sides is known.
In addition, primitive Pythagorean Triples are used in cryptography for the generation of keys.
