PYTHAGOREAN TRIPLE GENERATOR

About "Pythagorean Triple Generator"

Pythagorean Triple Generator : 

In this section, we will learn how to generate Pythagorean Triples. 

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 + n²  =  3² + 2²  =  9 + 4  =  13

Leg₁   =   m² - n²  =  3² - 2²  =  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 - Examples

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

After having gone through the stuff given above, we hope that the students would have understood, "Pythagorean Triple Generator". 

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