# PYTHAGOREAN TRIPLE GENERATOR

Let us consider the right triangle shown below.

We can use the following formulas to generate a Pythagorean triple.

Hypotenuse  =  m² + n²

Leg1  =  m² - n²

Leg2  =  2mn

In the above formulas, always m has to be greater than n.

That is,

m > n

In the above formulas, we can take any positive values for m and n to get the three sides of a right triangle such that m > n.

For example, let m  =  3 and  n  =  2.

Then, we have

Hypotenuse   =   m2 + n=  32 + 2=  9 + 4  =  13

Leg1   =   m2 - n2  =  32 - 2=  9 - 4  =  5

Leg2   =   2mn  =  2(3)(2)  =  12

Now, we can check whether 13, 5, 12 be the three sides of the right angle triangle using Pythagorean theorem.

That is,

(Hypotenuse)2   =   (Leg1)2 + (leg2)2

132   =   52 + 122

169  =  25 + 144

169  =  169

The three values 13, 5 and 12 satisfy the Pythagorean Theorem.

Therefore, the values 13, 5 and 12 form a Pythagorean triple.

## How to Get the Sides of a Right Triangle

Example 1 :

Using 5 and 6, create the lengths of three sides of a right triangle.

Solution :

Since m > n, we can take m  =  6 and n  =  5.

Then, we have

Hypotenuse   =   m2 + n2  =  62 + 52  =  36 + 25  =  61

Leg1   =   m2 - n2  =  62 - 52  =  36 - 25  =  11

Leg2   =   2mn  =  2(6)(5)  =  60

So, the lengths of three sides of the right triangle are

61,  11  and  60

Example 2 :

Using 4 and 3, create the lengths of three sides of a right triangle.

Solution :

Since m > n, we can take m  =  4 and n  =  3.

Then, we have

Hypotenuse   =   m2 + n2  =  42 + 32  =  16 + 9  =  25

Leg1   =   m2 - n2  =  42 - 32  =  16 - 9  =  7

Leg2   =   2mn  =  2(4)(3)  =  24

So, the lengths of three sides of the right triangle are

25,  7  and  24

Example 3 :

Using 1 and 2, create the lengths of three sides of a right triangle.

Solution :

Since m > n, we can take m  =  2 and n  =  1.

Then, we have

Hypotenuse   =   m2 + n2  =  22 + 12  =  4 + 1  =  5

Leg1   =   m2 - n2  =  22 - 12  =  4 - 1  =  3

Leg2   =   2mn  =  2(2)(1)  =  4

So, the lengths of three sides of the right triangle are

5,  3  and  4

Example 4 :

Using 6 and 7, create the lengths of three sides of a right triangle.

Solution :

Since m > n, we can take m  =  7 and n  =  6.

Then, we have

Hypotenuse   =   m2 + n2  =  72 + 62  =  49 + 36  =  85

Leg1   =   m2 - n2  =  72 - 62  =  49 - 36  =  13

Leg2   =   2mn  =  2(7)(6)  =  84

So, the lengths of three sides of the right triangle are

85,  13  and  84

Example 5 :

Using 11 and 6, create the lengths of three sides of a right triangle.

Solution :

Since m > n, we can take m  =  11 and n  =  6.

Then, we have

Hypotenuse   =   m2 + n2  =  112 + 62  =  121 + 36  =  157

Leg1   =   m2 - n2  =  112 - 62  =  121 - 36  =  85

Leg2   =   2mn  =  2(11)(6)  =  132

So, the lengths of three sides of the right triangle are

157,  85  and  132

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