Pythagorean triples are sets of three positive integers that satisfy the Pythagorean Theorem. This ancient theorem, attributed to the Greek mathematician Pythagoras, is fundamental in geometry and trigonometry. The theorem states that in any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The equation can be expressed as: a2 + b2 = c2
Where a and b are the lengths of the triangle's legs, and c is the length of the hypotenuse. Pythagorean triples are therefore the integer solutions to this equation.
In this article, we will learn about the Pythagorean triples, and their formulas along with some examples.
What are Pythagorean Triples?
Pythagorean triples are three positive integers that satisfy the Pythagoras theorem. Generally, these three terms can be written in the form (a, b, c), and form a right-angle triangle with c as its hypotenuse and a and b as its base and height. The triangle formed by these terms is known as the Pythagorean triangle.
Let us consider a right-angled triangle in which b is the base, a is perpendicular, and c is the hypotenuse. So, according to the Pythagoras theorem: the sum of squares of sides a and b is equal to the square of the third side c.
a2 + b2 = c2
Here, a, b, and c are base, perpendicular, and hypotenuse of right angle triangle.
Now in this case we say that a, b, and c are Pythagorean Triples.
Pythagorean Triples Examples
There are infinitely many possible Pythagorean triples as we can choose any two numbers for base and perpendicular and we can find hypotenuse using the Pythagoras theorem. For example, let's say the perpendicular of the triangle is 4 units, and the base is 3 units, then the hypotenuse will be 5 using the Pythagoras theorem. This is further explained in the image added below.
Apart from these, many other Pythagorean triplets can be generated with the help of these basic Pythagorean triples(i.e. 3, 4, 5). The best way to obtain more triples is to scale them up, as all the integral multiple of any Pythagorean triplet is also a Pythagorean triple i.e., as (3, 4, 5) is Pythagorean triple which implies that (3n, 4n, and 5n) is always a Pythagorean triple, where, n ∈ {1, 2, 3, 4, 5, . . . }. Below is the illustration of the same:
n | (3n, 4n, 5n) |
|---|
2 | (6, 8, 10) |
3 | (9, 12, 15) |
4 | (12, 16, 20) |
5 | (15, 20, 25) |
Common Pythagorean Triples
The most commonly used Pythagorean Triples are (3, 4, 5). Other than this there are more common examples such as (5, 12, 13), (6, 8, 10), (9, 12, 15), (7, 24, 25), and (15, 20, 25).
Types of Pythagorean Triples
Pythagorean Triples can further be classified into two types namely:
- Primitive Pythagorean Triples
- Non-Primitive Pythagorean Triples
Primitive Pythagorean Triples
Primitive Pythagoras triples are also known as Reduced triples. The greatest common factor of these triples is 1. Or we can say that primitive Pythagorean triples are those triples in which the three numbers do not have any common divisor other than one. Such type of triples only contains one even number among the three given numbers.
Example: 3, 4, 5 is a primitive Pythagorean triple.
As 3, 4, 5 satisfy the Pythagorus theorem and also the greatest common factor of (3, 4, 5) is 1.
Non-Primitive Pythagorean Triples
Non-primitive Pythagoras triples are also known as imprimitive Pythagorean triples. They are those triples in which the three numbers have a common divisor. Such types of triplets can contain more than one even positive number among the three given numbers.
Example: 6, 8, 10 is a Non-Primitive Pythagorean triple.
As, 6, 8, 10 satisfy the Pythagorean triples formula but the greatest common factor of 6, 8, 10 is 2 which is not equal to 1.
Pythagorean Triples Formula is derived from the right-angled triangle. The sides of the right-angle triangle arranged in increasing order as triples are Pythagorean triples. If two values out of three is given, the third can be obtained from the Pythagoras theorem which is also known as Pythagorean Triplets Formula.
The Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the base and perpendicular. Let's say the perpendicular is denoted by 'a', the base is denoted by 'b', and the hypotenuse is denoted by 'c', then the Pythagorean triples formula will be:
c2 = a2 + b2
Proof of the Pythagorean triplets Formula or Pythagoras theorem can be done in many ways. There are well over 371 proofs for this formula. Here we are using one of the many geometric methods. In this method, we use the figure as follows:
Step 1: We have four right-angled triangles with base m, perpendicular n, and hypotenuse p. Now arrange these triangles so that they make two squares one is outer square ABCD, whose side is m+n, and another one is inner square WZXY, whose side is p.
Step 2: Now we find the area of the inner square, outer square, and triangles:
Area of outer square, ABCD = (m + n)2
Area of inner square, WXYZ = (p)2
Area of one triangle = 1/2(m × n)
⇒ Area of four triangles (as all four triangles are the same) = 4 × 1/2(m × n)
⇒ Area of four triangles = 2(m × n)
Step 3: As we know that the area of square ABCD = Area of square WXYZ + Area of four triangles
⇒ (m + n)2 = 2(m × n) + p2
⇒ m2 + 2 × m × n + n2 = 2 × m × n + p2
⇒ m2 + n2 = p2
Hence ,Pythagorean triples Formula is proved.
The generation of Pythagorean triples is not a matter of mere chance; there is a systematic method to derive these triples using Euclid's formula. This formula states that given two positive integers m and n (where m > n > 0), a Pythagorean triple can be formed as follows:
a = m^2 - n^2, \, b = 2mn, \, c = m^2 + n^2
For example, let’s take m = 3 and n = 2. Plugging these values into Euclid's formula, we get the Pythagorean triple (5, 12, 13). This systematic approach makes it possible to generate an infinite number of Pythagorean triples.
Pythagorean triples are the positive integers and there are two cases for the number that can help us generate Pythagorean triples. The numbers can either be odd or even. The cases mentioned above can be explained in detail, as follows:
If Number is Odd
If the generator number (m) is Odd then the following formula can be used to find the other two numbers to form a triple. If we have an odd number as a generator i.e., 1, 3, 5, 7, 9, . . ., then using m the remaining two numbers of triple can be found by putting m in the formula.
[m, (m2 - 1)/2 , (m2 + 1)/2]
Note: Here m should be greater than 1.
Example: If m = 3, find the rest of the Pythagorean triplets.
Solution:
We have m =3, which is an odd number
Hence, we will put the value of m in [m, (m2 - 1)/2 , (m2 + 1)/2]
(m2 -1)/2 = (32 - 1)/2 = 8/2 = 4.
and (m2 + 1)/2 = (32 + 1)/2 = 10/2 = 5
Therefore, the Pythagorean triples are (3, 4, 5).
If Number is Even
If the generator number (m) is Even then we can use a different formula to find the other two numbers to form a triple. If we have an even number as a generator i.e., 2, 4, 6, 8,10, . . ., then using m the remaining two numbers of triple can be found by putting m in the following formula:
[m, (m2 - 4)/4, (m2 + 4)/4
Note: Here m should be greater than 2.
Example: Find the rest of the Pythagorean triples if m = 4.
Solution:
We have m = 4, which is an even number
Hence, we will put the value of [m, (m2 - 4)/4, (m2 + 4)/4]
(m2 - 4)/4 = (42 - 4)/4 = 12/4 = 3.
(m2 + 4)/4 = (42 + 4)/4 = 20/4 = 5.
Therefore, the Pythagorean triples are (3, 4, 5).
Note: Even if the method helps solve and find infinitely many Pythagorean triples, it still cannot find them all. For instance, the Pythagorean triples (20, 21, 29) cannot be formed using this technique. Thus, this formula is not absolute to find all possible Pythagorean triples.
Right Traingle Method to generate Pythagorean Triplets
Other than the method illustrated above in this article, there is another way to generate Pythagorean Triples. In order to generate Pythagorean triples, we can assume the sides of the right-angled triangle are a, b, and c and define these sides in terms of two integral values m and n, such that,
- a is the perpendicular of the triangle here, and a = 2mn.
- b is the base of the triangle, and b = m2 - n2.
- c is the hypotenuse of the triangle, and c = m2 + n2.
Note: Here, m and n are the co-prime numbers such that m>n.
These values of a and b in terms of m and n, clearly satisfy Pythagoras Theorem, which is shown as follows:
c2 = a2 + b2
⇒ (m2 + n2)2 = (2mn)2 + (m2 - n2)2
⇒ m4 + n4 + 2m2n2 = 4m2n2 + m4 + n4 - 2m2n2
⇒ m4 + n4 + 2m2n2 = m4 + n4 + 2m2n2
Thus, LHS = RHS,
Now simply use co-prime natural numbers in order to find the values of Pythagorean triplets. It is important to note that m must be greater than n.
Example: Find the Pythagorean triples when the values of m and n are 3 and 2, respectively.
Solution:
Pythagorean triples can be given as
- a = 2mn
- b = m2 - n2
- c = m2 + n2
Therefore, putting m = 3 and n = 2.
a = 2 × 3 × 2 = 12 units.
b = 32 - 22 = 9 - 4 = 5 units
c = 33 + 22 = 9 + 4 = 13 units.
Therefore, the Pythagorean triples are (5, 12, 13).
List of Common Pythagorean Triples
Below is the list of some of the Pythagorean triplets where the value of c (the hypotenuse of the triangle) is greater than 100:
| (20, 99, 101) | (60, 91, 109) | (15, 112, 113) | (44, 117, 125) |
| (88, 105, 137) | (17, 144, 145) | (24, 143, 145) | (51, 140, 149) |
| (85, 132, 157) | (119, 120, 169) | (52, 165, 173) | (19, 180, 181) |
| (57, 176, 185) | (104, 153, 185) | (95, 168, 193) | (28, 195, 197) |
| (84, 187, 205) | (133, 156, 205) | (21, 220, 221) | (140, 171, 221) |
| (60, 221, 229) | (105, 208, 233) | (120, 209, 241) | (32, 255, 257) |
| (23, 264, 265) | (96, 247, 265) | (69, 260, 269) | (115, 252, 277) |
These all values verify the Pythagorean triples formula a2 + b2 = c2.
Properties of Pythagorean Triples
For a right-angled triangle with base m, height n, and hypotenuse p, Pythagorean triples have the following properties:
- All Pythagorean triples are positive integers.
- Pythagorean triples can be represented as m, n, p, or (m, n, p).
- Pythagorean triples always satisfy the formula m2 + n2 = p2.
Fun Facts!
- There are infinite Pythagorean Triples.
- Pythagorean Triplet can either consist of all even numbers, or two odd numbers and one even number.
- All three numbers of a Pythagorean Triplet can never be odd.
Articles related to Pythagorean Triples:
Pythagorean Triples Solved Examples
Example 1: Find Pythagorean triples if m = 8.
Solution:
This is the case when the number is even:
Given m = 8,
So, (m2 - 4)/4 = (64 - 4)/4 = 15
and (m2 + 4)/4 = (64 + 4)/4 = 17
Hence Pythagorean triplets are 8, 15, 17.
Example 2: Find Pythagorean triples if m = 9.
Solution:
This is the case when the number is odd:
Given m = 9,
So, (m2 - 1)/2 = (81 - 1)/2 = 40
and (m2 + 1)/2 = (81 + 1)/2 = 41
Hence Pythagorean triples are 9, 40, 41.
Example 3: Find Pythagorean triples, one of whose members is 13.
Solution:
Take m = 13,
So, (m2 - 1)/2 = (169 - 1)/2 = 84
and (m2 + 1)/2 =(169 + 1)/2 = 85
Hence Pythagorean triples are 13, 84, 85.
Example 4: Check if (6, 8, 10) is a Pythagorean triplet or not.
Solution:
Let us take m = 6, n = 8, and p = 10
According to the formula
m2 + n2 = p2
⇒ (6)2 + (8)2 = (10)2
⇒ 36 + 64 = 100
⇒ 100 = 100
Here, L.H.S = R.H.S
Hence, (6, 8, 10) is a Pythagorean triple.
Example 5: If (y, 84, 85) is a Pythagorean triplet, then find the value of y.
Solution:
Let us take m = y, n = 84, and p = 85
According to the formula
m2 + n2 = p2
⇒ (y)2 + (84)2 = (85)2
⇒ y2 + 7056 = 7225
⇒ y2 = 169
⇒ y = 13 [as y can't be negative]
Practice Questions on Pythagorean Triples
Question 1: Find the Pythogorean triples of 21.
Question 2: Prove that (12, 35, 37) is a Pythagorean triple.
Question 3: Find x, if (11, x, 61) is a Pythagorean triple.
Question 4: Find the other two numbers of a Pythagorean triple, if one of the number is 5.
Question 5: Check if (4, 7, 9) is a Pythagorean triple or not.
Conclusion
The article discusses Pythagorean triples, which are sets of three integers (a, b, c) that satisfy the equation a2 + b2 = c2 and represent the sides of a right triangle. It explains how these triples can be generated using formulas based on whether the initial integer (m) is odd or even, offering specific examples such as (3, 4, 5) and (5, 12, 13). The article also covers methods for creating these triples by manipulating two co-prime integers, m and n, ensuring that m is greater than n, to satisfy the Pythagorean theorem. This exploration of Pythagorean triples illustrates their fundamental role in geometric and algebraic concepts, emphasizing their practical and theoretical significance in mathematics.
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