Pythagorean triples are a^{2}+b^{2} = c^{2} 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 90^{0 }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.
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What are Pythagorean Triples?
The integer solutions to the Pythagorean Theorem, a^{2} + b^{2} = c^{2} are called Pythagorean Triples which contains three positive integers a, b, and c.
Example: (3, 4, 5)
By evaluating we get:
3^{2} + 4^{2} = 5^{2}
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
(3, 4, 5) | (5, 12, 13) | (8, 15, 17) | (7, 24, 25) |
(20, 21, 29) | (12, 35, 37) | (9, 40, 41) | (28, 45, 53) |
(11, 60, 61) | (16, 63, 65) | (33, 56, 65) | (48, 55, 73) |
(13, 84, 85) | (36, 77, 85) | (39, 80, 89) | (65, 72, 97) |
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:
p^{2} + q^{2} = r^{2}.
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:
(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) |
Students can pick any triples from the above list and prove the Pythagoras formula,i.e.,
a^{2}+b^{2}=c^{2}
How to Find Pythagorean Triples?
These rules are as below:
- Every odd number is the p side of a Pythagorean triplet.
- The q side of a Pythagorean triplet is simply (p^{2}– 1)/2.
- The r side is (q^{2} + 1)/2.
Now, p and r are always odd; q is even.
- These relationships are true as the difference between successive square numbers is successive odd numbers.
- All odd numbers which are itself a square (and the square of all odd number is an odd number itself) thus giving a Pythagorean triplet.
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 = b^{2}.
The area of the entire square = 4(1/2(ab)) + c^{2}
Now we can conclude that
(a + b)^{2} = 4(1/2 (ab)) + c^{2}.
or
a^{2} + 2ab + b^{2} = 2ab + c^{2}.
Simplifying, we get Pythagorean triples formula,
a^{2} + b^{2} = c^{2}
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
The common Pythagorean triples are given here in the table:
Pythagorean Triples | x 2 (Times 2) | x 3 (Times 3) | x 4 (Times 4) |
3-4-5 | 6-8-10 | 9-12-15 | 12-16-20 |
5-12-13 | 10-24-26 | 15-36-39 | 20-48-52 |
7-24-25 | 14-48-50 | 21-72-75 | 28-96-100 |
9-40-41 | 18-80-82 | 27-120-123 | 36-160-164 |
11-60-61 | 22-120-122 | 33-180-183 | 44-240-244 |
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.
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Video Lesson
Frequently Asked Questions – FAQs
What are Pythagorean triples?
What are the five most common Pythagorean triples?
(5,12,13)
(7,24,25)
(9,40,41)
(11,60,61)
How to find Pythagorean triplets?
Every odd number is the p side of a Pythagorean triplet.
The q side of a Pythagorean triplet is simply (p^{2}– 1)/2.
The r side is (q^{2} + 1)/2.
If p=9
q=(9^{2}-1)/2 = (81-1)/2 = 80/2 = 40
r=(9^{2}+1)/2 = (81+1)/2 = 82/2 = 41
Hence, (9,40,41) are the Pythagoras triples.
How to do scaling of triples?
(6,8,10)
So, 6^{2}+8^{2}=10^{2}
36+64=100
100 = 100
Is (4,5,8) is Pythagorean triple?
Let us take LHS first,
4^{2}+5^{2} = 16+25 = 41
RHS = 82 = 64
Clearly, 41 is not equal to 64
Therefore (4,5,8) is not a pythagorean triples.