# mat 126 rewritten tonight

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