In any triangle we know that the sum of all the angles is 180o which we call as angle sum property. Hence, in any triangle if we are given the measures of two angles we will be able to find the third angle using this angle sum property. The properties of angles is such that, the sides opposite to a vertex depend on the vertex angle. The side opposite to the greatest angle of a triangle is greater. Similarly the side opposite to the smaller angle of a triangle is smaller. Based on this we shall discuss with the possible sides of a right triangle when the length of the sides is a positive integer (whole number).

## Pythagorean Triple Definition

Right Angled triangle: A triangle in which one of the angle measure 90o, is called a right angled triangle. The side opposite to 90o of a right triangle is called its hypotenuse. The sides containing the right angle are called its legs.

The following diagram shows a right angled triangle.

Pythagoras Theorem: According to this theorem, in a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other  two sides (legs).

The following triangle represents a triangle which satisfy Pythagoras theorem.

In the above figure, Area of the square on the side AC = Area of the square on the side AB + Area of the square on the side BC                                                    => AC2  = AB2 + BC2
Pythagorean Triple Definition: A set of three positive integers which satisfy the condition, that the square if the largest is equal to the sum of the squares of the other two sides.

We can also state that length of the sides of the right triangle (in integers) which satisfy the Pythagoras theorem, is called a Pythagorean Triple.
For example, consider, the whole numbers 3, 4 and 5
we see that 5 > 4 > 3
(i.e) 5 is the largest.
Therefore according to the above definition, 5= 32 + 42
=>             25 = 9 + 16
=>             25 = 25
Therefore, the positive integers 3, 4 and 5 are called as Pythagorean Triples.

## Pythagorean Triples Formula

If m and n are positive integers > 1 and m > n, then the triangle whose sides are given by 2mn, m2 - n2 and m2 + n2 form a right angle triangle.
(i.e), ( m2 + n2 ) = ( m2 - n2 )2 + ( 2 m n )2
=> ( m2 + n2 )2 = m4 - 2 m2 n2 + n2 + 4m2 n2
=> n4 + 2 m2 n2 + 1 = n4 + 2m2 n2 + 1
=> both sides are equal.

### Common Pythagorean Triples:

Substituting values for m and n we get the following triples.

 Serial No. m n 2mn m2 - n2 m2 + n2 Pythagorean Triples 1. 2 1 4 4 - 1 = 3 4 + 1 = 5 (3, 4, 5) 2. 3 1 6 9 - 1 = 8 9 + 1 = 10 (6, 8, 10) 3. 3 2 12 9 - 4 = 5 9 + 4 = 13 (5, 12, 13) 4. 4 1 8 16 - 1 = 15 16 + 1 = 17 (8, 15, 17) 5. 4 2 16 16 - 4 = 12 16 + 4 = 20 (12, 16, 20) 8. 4 3 24 16 - 9 = 7 16 + 9 = 25 (7, 24, 25)

Primitive Pythagorean Triples: The Pythagorean triples, a, b, c in which gcd (a, b, c) = 1 are called Primitive Pythagorean Triples and the corresponding right triangle is called a Primitive Right Triangle.

From the above table, Primitive Pythagorean triples are,
(3, 4, 5), (5, 12, 13), (8, 15, 17) and (7, 24, 25)

## Pythagorean Triples Examples

### Solved Examples

Question 1: Find the Pythagorean triples, one of whose value is 14. Also verify if the Pythagorean Triple is a Primitive Pythagorean Triples.
Solution:

We are given that one of the side = 14 units
Let 2 m n = 14
=> m n = $\frac{14}{2}$
= 7

Since m and n are positive integers and m > 1, we have n = 1 [the smallest positive integer]
Therefore, we have  m ( 1 ) = 7
=>                               m = 7
Therefore,            m2 - n2  = 72 - 12
= 49 - 1
= 48
m2 +  n2   = 72 + 12
= 49 + 1
= 50
The required Pythagorean Triple is (14, 48, 50).
since gcd (14, 48, 50) = 2 $\neq$ 1, we see that the triples are not of Primitive Pythagorean Triples.

Question 2: Find the Pythagorean Triples, one of whose value is 12. Also verify if the Pythagorean Triple is a Primitive Pythagorean Triples.
Solution:

We are given that one of the value = 12
Let 2 m n = 12
=>       m n = $\frac{12}{2}$
= 6
Let n = 1, Therefore, m = 6
Therefore, m2 - n= 62 - 12
= 36 - 1
= 35
m2 + n2  = 62 + 12
= 36 + 1 = 37
The corresponding Pythagorean Triple is (12, 35, 37)
Let n = 2, Therefore m ( 2 ) = 6
=>     2 m = 6
=>        m = $\frac{6}{2}$
= 3
Therefore substituting in the above Pythagorean Triples Formula, we get,
m2 - n= 32 - 22
= 9 - 4
= 5
m2 +  n2 = 32 + 22
= 9 + 4
= 13
The corresponding Pythagorean Triplet is (5, 12, 13)

Since g c d (12, 35, 37) = 1 and
g c d (5, 12, 13) = 1
we find that the triples are of Primitive Pythagorean Triples.