Table of Contents

Last modified on August 3rd, 2023

Pythagorean Triples are a set of 3 positive integers, namely a, b, and c that perfectly satisfy the Pythagorean Theorem rule:

**a ^{2} + b^{2} = c^{2}**, 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 a^{2} + b^{2} = c^{2}

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

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,

a^{2} + b^{2}

=> 3^{2} + 4^{2}

=> 9 + 16

=> 25

Again,

c^{2} => 5^{2} => 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).

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,

a^{2} + b^{2}

=> 6^{2} + 8^{2}

=> 36 + 64

=> 100

Again,

c^{2} => 10^{2} => 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) |

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:

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 = m^{2} – n^{2}

^{ }= 2^{2} – 1^{2}

= 4 -1

= 3

**Solving for b**

As we know,

b = 2mn

= 2 × 2 × 1

= 4

**Solving for c**

As we know,

c = m^{2} + n^{2}

= 2^{2} + 1^{2}

= 4 +1

= 5

As we know, (3, 4, 5) is the smallest Pythagorean Triple, and thus satisfy the Pythagorean Theorem rule

**a ^{2} + b^{2} = c^{2}**

**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:

a^{2} + b^{2} = c^{2}

=> 6^{2} + 8^{2 }= 10^{2}

=> 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:

a^{2} + b^{2} = c^{2}

=> 9^{2} + 12^{2 }= 15^{2}

=> 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) = [(m^{2} – n^{2}), (2mn), (m^{2} + n^{2})], here c = 25 cm

Thus,

m^{2} + n^{2} = 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 = m^{2} – n^{2}

= 5^{2} – 3^{2}

= 25 -9

= 16**Solving for b**

As we know,

b = 2mn

= 2 × 5 × 3

= 30**Solving for c**

As we know,

c = m^{2} + n^{2}

= 5^{2} + 3^{2}

= 25 +9

= 34

As we know, to be a Pythagorean triple, it must satisfy the Pythagorean Theorem rule:

a^{2} + b^{2} = c^{2}

=> 16^{2} + 30^{2} = 34^{2}

=> 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:

a^{2} + b^{2} = c^{2}, here b = 35, c = 37

=> a^{2} + 35^{2} = 37^{2}

=> a^{2} = 37^{2} – 35^{2}

=> a^{2} = 144

=> a = 12

Thus, the value of ‘a’ in the Pythagorean Triple (a, 35, 37) is 12

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.

Last modified on August 3rd, 2023