The knowledge of Pythagorean theorem is a prerequisite to understand 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 Pythagorean Theorem by several Physicists and Mathematicians. There are manifold approaches for arriving at the proof of the Pythagorean theorem.
A triple refers to the number ‘3’. Pythagorean triples refer to the sets of 3 numbers (mostly integers) which satisfy the rule of Pythagorean theorem. Just to recall, the Pythagorean theorem relates the squares on the sides of a right triangle. It states that, in 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 has 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 Pythagoras theorem. It is most common to represent the Pythagorean triples as three alphabets (a, b, c) which represents 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’ represents 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:
In the above equation, ‘c’ represents the largest number among the three numbers.
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 Pyhagorean rule and also have a common divisor.
This triplet may or may not have more than one even positive integers.
Example for non primitive Pythagorean triples is 6, 8, 10.
If ‘a’, ‘b’ and ‘c’ are the three sides of a right triangle, ‘c’ being the longest side, it can be written that a2 + b2 = c2 according to the Pythagoras theorem. It can be mathematically derived that the measure of ‘a’, ‘b’ and ‘c’ can be determined using the formula:
These formulas can be used to find any number of Pythagorean triples by substituting the values of ‘m’ as any natural number.
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.
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.
1. Find the Pythagorean triplet that consists of 18 as one of its elements.
Let us consider the Pythagorean triplet (a, b, c) in which
a = m2 - 1, b = 2 m 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).
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
1. What is a Pythagorean Triplet? Give Examples.
Answer: 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 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 to 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 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.