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Pythagorean Triples - 3² + 4² = 5²











































complete the exercise 4 in the “Projects” section on page 522 of Mathematics in Our World.

The numbers 3, 4, and 5 are called Pythagorean triples
since 3
2
+ 4
2
= 5
2
The numbers 5, 12, and 13 are also .
Pythagorean triples since 5
2
+ 12
2
= 13
2
Can you find .
any other Pythagorean triples? Actually, there is a set
of formulas that will generate an infinite number of
Pythagorean triples. Research the topic of Pythagorean
triples and write a brief report on the subject.

Make sure you build or generate at least five more Pythagorean Triples using one of the many formulas available online for doing this.
After building your triples, verify each of them in the Pythagorean Theorem equation.

Comments for Pythagorean Triples - 3² + 4² = 5²

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Apr 15, 2011
Pythagorean Triples
by: Staff


The question:

Complete the exercise 4 in the “Projects” section on page 522 of Mathematics in Our World.

The numbers 3, 4, and 5 are called Pythagorean triples since 3² + 4² = 5².

The numbers 5, 12, and 13 are also Pythagorean triples since 5² + 12² = 13².


Can you find any other Pythagorean triples?

Actually, there is a set of formulas that will generate an infinite number of Pythagorean triples.

Research the topic of Pythagorean triples and write a brief report on the subject.

Make sure you build or generate at least five more Pythagorean Triples using one of the many formulas available online for doing this.

After building your triples, verify each of them in the Pythagorean Theorem equation.


The answer:

A PYTHAGOREAN TRIPLE is simply a RIGHT TRIANGLE whose SIDES are positive INTEGERS.

An easy 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:

Choose ANY TWO 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


The formula shown in the last example will not give all possible Pythagorean Triples.

To calculate all 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





Thanks for writing.


Staff
www.solving-math-problems.com



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