Pythagorean Triples

Pythagorean triples are a²+b² = c² where a, b and c are the three positive integers. These triples are represented as (a,b,c). Here, a is the perpendicular, b is the base and c is the hypotenuse of the right-angled triangle. The most known and smallest triplets are (3,4,5). Learn Pythagoras theorem for more details.

Pythagoras who was a mathematician was interested in mathematics, science, and philosophy. He was born in Greece in about 570 BC. He is famous for a property of triangles with a right angle i.e 90⁰ angles, and the property is known as Pythagoras Theorem. In a right-angled triangle, the hypotenuse is the side ‘r’, the side opposite the right angle. Adjacent to the right angle the shorter of the two sides is the side p. In this article, let us discuss what is Pythagorean triples, its formula, list, steps to find the triples, examples, and proof.

What are Pythagorean Triples?

The integer solutions to the Pythagorean Theorem, a² + b² = c² are called Pythagorean Triples which contains three positive integers a, b, and c.

Example: (3, 4, 5)

By evaluating we get:

3² + 4² = 5²

9+16 = 25

Hence, 3,4 and 5 are the Pythagorean triples.

You can say “triplets,” but “triples” are the favoured term. Let’s start this topic by an introduction of Pythagoras theorem.

Table

Pythagoras Triples Formula

If a triangle has one angle which is a right-angle (i.e. 90^o), there exists a relationship between the three sides of the triangle.

If the longest side (called the hypotenuse) is r and the other two sides (next to the right angle) is called p and q, then:

p² + q² = r².

or,

The sum of the squares of the other two sides is the same as the square of the longest side.

Pythagorean Triples List

The list of Pythagorean triples where the value of c is above 100 is given below:

Students can pick any triples from the above list and prove the Pythagoras formula,i.e.,

a²+b²=c²

How to Form Pythagorean Triples?

How to Form Pythagorean Triples?

As we know, the number can be an odd number or an even number. Now, let us discuss how to create the Pythagorean triples.

Case 1: If the number is odd:

Let us assume the number be “x”.

If “x” is odd, then the Pythagorean triple = x, (x²/2) – 0.5, (x²/2) + 0.5.

Consider an example (7, 24, 25). Now, let us discuss how to form this Pythagorean triple.

Hre, x = 7, which is an odd number.

(x²/2) – 0.5 = (49/2) – 0.5 = 24.5 – 0.5 = 24

(x²/2) + 0.5 = (49/2) + 0.5 = 24.5 + 0.5 = 25

Hence, the Pythagorean triple formed is (7, 24, 25).

Case 2: If the number is even:

If “x” is even, then the Pythagorean triple = x, (x/2)²-1, (x/2)²+1.

Let us assume an example, (16, 63, 65). Now, we will check how to form the Pythagorean triple.

Here, x = 16, which is an even number.

(x/2)² -1 = (16/2)² – 1 = 8² – 1 = 64 – 1 = 63

(x/2)² +1 = (16/2)² +1 = 8² + 1 = 64 + 1 = 65.

Thus, the Pythagorean triple formed is (16, 63, 65)

Note: We can observe that there are infinitely many Pythagorean triples because the approaches give us a triple for every positive whole number. Can these procedures, however, produce all of them? The answer is “No”. For example, the Pythagorean triple (20, 21, 29) cannot be formed using these methods.

Try out: Pythagorean Triples Calculator

Pythagorean Triples Proof

Proof of Pythagoras theorem:

Look at the figure above

In the figure, at left,

Area of square = (a+b)²

Area of Triangle = 1/2(ab)

Area of the inner square = b².

The area of the entire square = 4(1/2(ab)) + c²

Now we can conclude that

(a + b)² = 4(1/2 (ab)) + c².

or

a² + 2ab + b² = 2ab + c².

Simplifying, we get Pythagorean triples formula,

a² + b² = c²

Hence Proved.

Triangular Numbers

The difference between successive squares is successive odd numbers is a fact and suggests that every square is the sum of two successive triangular numbers.

And in this, the triangular numbers are the successive sums of all integers.

• 0 + 1 = 1,
• 0 + 1 + 2 = 3,
• 0 + 1 + 2 + 3 = 6, etc.

So the triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, etc.

105 + 120 = 225; 225 is the square of 15.

Common Pythagorean Triples

As we know, the specific set of integers that satisfies the Pythagoras theorem is called Pythagorean triples. It means that the set of integer numbers has a special connection with the Pythagoras theorem. Not only the set satisfies the Pythagoras theorem, but also the multiples of the integer set also satisfy the Pythagoras theorem.

For example, (3, 4, 5) is the most common Pythagorean triples. When each integer number is multiplied by 2, we get the set (6, 8, 10), which also satisfies the Pythagoras theorem.

(i.e.,) 3² + 4²  = 5²

9+16 = 25

25 = 25

Similarly,6² + 8² = 10²

36 + 64 = 100

100 = 100

This can be simply expressed as follows:

If a, b and c are the positive integers, which satisfies the Pythagoras theorem, then ak, bk, ck will also satisfy the Pythagoras theorem, when “k” is a positive integer.

Also, the Pythagorean triples can be found using different methods, such as generalized Fibonacci sequence, quadratic equations, using matrices and linear transformations, and so on. The set of Pythagorean triples is endless. We can prove that we have infinitely many Pythagorean triples with the help of (3, 4, 5)

Facts: An interesting fact about Pythagorean triples is that Pythagorean triples always consist of all even numbers or two odd numbers and an even number. 

A Pythagorean triple never be made up of all odd numbers or two even numbers and an odd numbers

Pythagorean Triples Examples (With Answers)

• So, the square of 3, 9, is the difference between 16, the square of 4, and 25 the square of 5, giving us the triplet 7,24,25.
• Similarly, the square of 5, 25 is the difference between 144, the square of 12, and 169, the square of 13, giving us the triplet 5, 12, 13.

Pythagorean Triples Problems

Example 1: 

Prove that (5, 12, 13) is a Pythagorean triple?

Solution:

To prove: (5, 12, 13) is a Pythagorean Triple

We know that, a² + b²  = c² 

(a, b, c) = (5, 12, 13)

Now, substitute the values, 

5² + 12² = 13²

25 + 144 = 169

169 = 169

Hence, the given set of integers satisfies the Pythagoras theorem, (5, 12, 13) is a Pythagorean triples.

Example 2: 

Check if (7, 15, 17) are Pythagorean triples.

Solution:

(a, b, c) = (7, 15, 17)

We know that a² + b²  = c² 

By substituting the values in the equation, we get

7² + 15² = 17²

49 + 225 =  289

274 ≠ 289

Hence, the given set of integers does not satisfy the Pythagoras theorem, (7, 15, 17) is not a Pythagorean triplet. Also, it proves that the Pythagorean triples are not made up of all odd numbers.

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