Introduction to Pythagorean triples
The knowledge of the Pythagorean theorem is a prerequisite to understanding the Pythagorean triples formula. The Pythagorean theorem states that the square on the longest side of a right triangle is equal to the sum of the squares on the other two sides of it. In a right triangle, the longest side is called the hypotenuse.
The theorem was first stated and proved by the Greek Mathematician Pythagoras and hence his name was given to the theorem as a token of appreciation. Many advanced research works and Mathematical concepts were explained based on the Pythagorean Theorem by several Physicists and Mathematicians. There are manifold approaches for arriving at the proof of the Pythagorean theorem.
What is the Pythagorean Triples Formula?
A triple refers to the number '3'. Pythagorean triples refer to the sets of 3 numbers (mostly integers) which satisfy the rule of the Pythagorean theorem. Just to recall, the Pythagorean theorem relates the squares on the sides of a right triangle. It states that, in the case of a right triangle, the square on the longest side has an area equal to the sum of the areas of the squares on the other two sides (the base and the perpendicular). Pythagorean triples have a set of three integers (mostly positive) such that the square of the largest among the three numbers is equal to the sum of the squares of the other two integers.
The Pythagorean triples formula has three positive integers that abide by the rule of the Pythagoras theorem. It is most common to represent the Pythagorean triples as three alphabets (a, b, c), which represent the three sides of a triangle. The right triangles constructed with the sides' a', 'b' and 'c' are called Pythagorean triangles. In this set of 3 integers, 'a' and 'b' represent the base and perpendicular, whereas 'c' represents the measure of the hypotenuse. So, according to the Pythagorean triples formula, it is a set of 3 positive integers such that:
a2+b2=c2
In the above equation, ‘c’ represents the largest number among the three numbers.
Pythagorean Triples Proof
Proof of Pythagoras theorem:
(Image will be Updated soon)
The figure above illustrates this
In the figure, at left,
Area of square = (a+b)2
Area of Triangle =12(ab)
Area of the inner square = b2.
The area of the entire square = 4(12(ab)) + c2
Therefore, it can be concluded that
(a + b)2 = 4(12 (ab)) + c2.
Or
a2 + 2ab + b2 = 2ab + c2.
Using Pythagorean triples, we get:
a2 + b2 = c2
Hence Proved.
Mathematics of Pythagorean Triples Formula
Using the following formula, we can find the Pythagorean triples. In the case of a triangle whose sides are a, b, and c, the same formula can be used to determine a, b, and c.
a =m2 – 1
b = 2m
c = m2 – 1
This results in a right-angled triangle with sides a, b, and c. These formulas can be used to find any number of Pythagorean triples by substituting the values of ‘m’ as any natural number.
For example:
If m = 2,
a = m2 - 1 = 22 - 1 = 4 - 1 = 3
b = 2 m = 2 x 2 = 4
c = m2 + 1 = 22 + 1 = 4 + 1 = 5
So, (3, 4, 5) is a Pythagorean triple.
Note:
In the formula for Pythagorean triples, the value of ‘m’ cannot be 0 and 1 because the sides of a triangle cannot be ‘0’ units.
Types of Pythagorean Triples
Pythagorean triples are of two types namely Primitive Pythagorean Triples and Non Primitive Pythagorean triples.
Primitive Pythagorean Triples
The set of three positive integers satisfying Pythagoras theorem is said to be a primitive Pythagorean triplet if and only if all the three numbers in the triples do not have any common divisor other than one.
A primitive Pythagorean triplet can have only one even positive integer among all the three.
Example: (3, 4, 5) is a primitive Pythagorean triplet.
Non Primitive Pythagorean Triples
A non primitive Pythagorean triplet is a set of 3 positive integers which adhere to the Pythagorean rule and also have a common divisor.
This triplet may or may not have more than one even positive integer.
Examples for non primitive Pythagorean triples are 6, 8, 10.
More details about Pythagorean triples
The Pythagorean triples are the specific set of integers that satisfy the Pythagorean theorem. This implies that the set of integer numbers has a special relationship with Pythagoras's theorem. Pythagoras' theorem is satisfiable not only by the set but also by the multiples of the integer set.
The most common Pythagorean triple is (3, 4, 5). We get the set (6, 8, 10) when we multiply each integer number by 2, which also satisfies Pythagoras' theorem.
(i.e.,) 32 + 42 = 52
9+16 = 25
25 = 25
Similarly,62 + 82 = 102
36 + 64 = 100
100 = 100
In simple terms, it is as follows:
If a, b, and c are positive integers, which satisfy the Pythagoras theorem, then ak, bk, ck will also satisfy the Pythagoras theorem if "k" is a positive integer.
We can see such Pythagorean triplets in the table below:
Interesting fact about Pythagorean triples:
You can observe that each Pythagorean triples consist of either all even numbers or two odd numbers. A Pythagorean triplet can never have two even numbers and one odd number or with all three odd numbers.
Below is a list of Pythagorean triples where c is greater than 100:
In addition to the above list, students can pick any triple and prove Pythagoras' formula, i.e. a2+b2=c2
Example Problems on Pythagorean triples
Example 1: Can you prove that (5, 12, 13) is a Pythagorean triple?
Solution:
In order to show that (5, 12, 13) is a Pythagorean triple
In mathematics, a2 + b2 = c2
(a, b, c) = (5, 12, 13)
Substitute the values now,
52 + 122 = 132
25 + 144 = 169
169 = 169
Therefore, the given set of integers satisfied Pythagoras's theorem: (5, 12, 13) is a Pythagorean triple.
Example 2: Check whether (7, 15, 17) are Pythagorean triples.
Solution:
(a, b, c) = (7, 15, 17)
In mathematics, a2 + b2 = c2
When we substitute the values in the equation, we get
72 + 152 = 172
49 + 225 = 289
274 ≠ 289
The given set of integers does not satisfy the Pythagorean triplet theorem; (7, 15, 17) is not a Pythagorean triplet. In addition, this proves that the Pythagorean triples do not include all odd numbers.
Example 3: Find the Pythagorean triplet that consists of 18 as one of its elements.
Solution:
Let us consider the Pythagorean triplet (a, b, c) in which
a = m2 - 1, b = 2m and c = m2 + 1
Let us take the value of ‘b’ as 18.
b = 2m
18 = 2m
m = 9
Substituting m = 9 in the formulas for ‘a’ and ‘c’, we get
a = m2 - 1 = 92 - 1 = 81 - 1 = 80
c = m2 + 1 = 92 + 1 = 81 + 1 = 82
Therefore, the Pythagorean triplet is (80, 18, 82).
Fun Quiz
1. Identify whether the given set of numbers is a Pythagorean triplet example or not.
7, 12, 13
3, 4, 5
2, 3, 7
6, 8, 10
2. Classify the following as primitive and non primitive Pythagorean triples.
5, 12, 13
6, 8, 10
9, 40, 41
15, 112, 113
14, 48, 50
FAQs on Formula for Pythagorean triples
1. What is a Pythagorean Triplet? Give Examples.
A Pythagorean triplet is a set of 3 positive integers which satisfies the Pythagorean theorem. In general, the three positive integers of a Pythagorean triplet are represented by the lower case letters of the English alphabet (a, b and c). The letter ‘c’ represents the largest number whose square is equal to the sum of the squares of the other two numbers. This is mathematically represented as: c2 = a2 + b2.
A triangle whose sides can be represented as a Pythagorean triplet is called a Pythagorean triangle. It should obviously be a right triangle because it satisfies the rule of the Pythagoras theorem.
2. What are the Practical Applications of Pythagorean Triples?
No three odd numbers can form a Pythagorean triplet. A Pythagorean triplet either consists of two odd and an even number or all three even numbers.
If all the three numbers of a Pythagorean triplet are multiplied by a constant, the resulting set of 3 numbers is also a Pythagorean Triplet.
A Pythagorean triplet can never have 2 even numbers and one odd number.
Adhering to a Mathematical fact, the sum of two even numbers is always even and the sum of an odd and an even number is odd,
The value of ‘c’ is odd if ‘a’ is even and ‘b’ is odd (or vice versa).
The value of ‘c’ is even if the values of both ‘a’ and ‘b’ are even.
3. What are the five most common Pythagorean triples?
Five most common Pythagorean triples are:
(3,4,5)
(5,12,13)
(6,8,10)
(7,24,25)
(9,40,41)
4. How do you scale triples?
In the case of (3,4,5) being Pythagorean triples, by scaling them by 2, we get:
(6,8,10)
So, 82+62=102
64+36=100
100 = 100
