Name

Math 126 Survey of Mathematical Methods

Pythagorean Triple

Instructor

Date

When we began to deal with Pythagorean Triples it can be very hard and difficult in doing any kind of mathematics problems. When we first starting this class and saw that it involve doing Pythagorean Triples knew it was going to be a challenge. When we deal with using formula you must know how to use them in the proper order and make sure you are using the correct one as well. If we can do that then we can be good at doing the problems as well as use the formulas later on.

A Pythagorean triple is simply a right triangle whose sides are positive integers.

After reviewing here’s the way to generate Pythagorean Triples is to multiply any known Pythagorean Triple by an integer (any integer).

Sides of a known triple: 3,4,5

Multiply by 2 = 6,8,10

Verification: 6² + 8² = 10² = 100

Multiply by 3 = 9,12,15

verification: 9² + 12² = 15² = 225

Multiply by 4 = 12,16,20

verification: 12² + 16² = 20² = 400

Sides of a known triple: 5,12,13

Multiply by 2 = 10,24,26

verification: 10² + 24² = 26² = 676

Multiply by 3 = 15,36,39

verification: 15² + 36² = 39² = 1521

Multiply by 4 = 20,48,52

verification: 20² + 48² = 52² = 2704

Sides of a known triple: 7,24,25

Multiply by 2 = 14,48,50

verification: 14² + 48² = 50² = 2500

Multiply by 3 = 21,72,75

verification: 21² + 72² = 75² = 5625

Multiply by 4 = 28,96,100

verification: 28² + 96² = 100² = 10000

In addition, there are many formulas

A Pythagorean Triple (a² + b² = c²) can be calculated using the following method:

By choosing any tow integers: x and y. y must be greater than x.

The sides of a new Pythagorean Triple are:

a = 2*x*y, b = y² – x², and c = y² + x²

for example, let x = 5 and y = 6

a = 2*x*y = 2*5*6 = 60

b = y² – x² = 6² – 5² = 36 – 25 = 11

c = y² + x² = 6² + 5² = 36 + 25 = 61

the sides of the new Pythagorean Triple are: 60,11,61

verification: 60² + 11² = 61² = 3721

Here’s how I calculate a possible Pythagorean Triples, use the following formula:

a = 2*d*x*y

b = d*(y^2 – x^2)

c = d*(y^2 + x^2)

d = any positive integer

y > x > 0

x and y must be positive integers

x and y must be even, odd; or odd, even integers

In conclusion if you were to ever have to use this formula again we will be able to do so and be able to user it in the correct manner as well. Pythagorean Triples is one of those formulas that if you use it once you will never forget it when you see it again. Out of all the formulas that have been taught in this class this is one that I will always remember.

References

Bluman, A. G. (2005). Mathematics in our world (1st ed. Ashford University Custom). United States: McGraw-Hill.