Pythagorean Triples

Pythagorean Theorem

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)

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

Pythagoras Triples Formula

If a triangle has one angle which is a right-angle (i.e. 90o), 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.

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 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.

Pythagorean Triples list

(3, 4, 5) (5, 12, 13) (7, 24, 25) (8, 15, 17)
(9, 40, 41) (11, 60, 61) (12, 35, 37) (13, 84, 85)
(16, 63, 65) (20, 21, 29) (28, 45, 53) (33, 56, 65)
(36, 77, 85) (39, 80, 89) (48, 55, 73) (65, 72, 97)

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 (p2 – 1)/2.
  • The r side is q + 1.

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.

Pythagorean Triples Examples

  • 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 Proof:

Proof of Pythagoras theorem:

Pythagorean Triples

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 + b2.

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.


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