What are Pythagorean Triads?

Question

I'm doing a maths assignment and one of the questions is to find a method of developing Pythagorean Triads but I dont know what to do! Help!

Answer 1

Pythagorean Triads (or Triples) are 3 integers that can form the sides of a right triangle. They are *integers* that will satisfy:

a² + b² = c²



An couple examples are:

(a,b,c) = (3,4,5)

or = (5,12,13)



There's an easy way to generate Pythagorean Triples. Choose two positive integers (one odd and one even) where m > n, and where m and n have no factors in common.



Plug them into the following 3 expressions:

m² - n²

2mn

m² + n²



Those numbers will form a primative Pythagorean Triple. Here's an example:

m = 2, n = 1



m² - n² = 2² - 1² = 3

2mn = 2(2*1) = 4

m² + n² = 2² + 1² = 5



Notice: You easily form another triple from an existing primative triple, by just multiplying all the numbers by a constant. Examples would be 6, 8, 10 (double) or 9,12,15 (triple). These are also Pythagorean Triples (Triads), but just not "primative" triples.

Answer 2

A Pythagorean triad (or "Pythagorean triple") consists of three positive integers (A, B, C) such that:



A^2 + B^2 = C^2



For a Pythagorean triad (A, B, C), if you construct a triangle of sidelengths A, B and C, it will be a right triangle, and C will be its hypotenuse.



The 3-4-5 triangle is a well-known right triangle with integer-length sides, and also forms the smallest possible Pythagorean triad (3, 4, 5).



--



You could list the set of perfect squares {1, 4, 9, 16, 25, 36, ... } and then add pairs at random until you find a pair that adds up to a perfect square. But this would be tedious.



But, if we know some triad for (A, B, C) such that A^2 + B^2 = C^2, what if we multiply each side of the equation by some value n^2 ? Then we'd get:



n^2 A^2 + n^2 B^2 = n^2 C^2



(nA)^2 + (nB)^2 = (nC)^2



Now we have a new Pythagorean triad: (nA, nB, nC).



So, actually, this is the same as if we had simply multiplied A, B and C by n, (not necessarily n^2)--which means we have found a method of finding new triads using any value of n:



--



Starting with (3, 4, 5), try multiplying each value by some n ... say, 2.



This results in the triad (6, 8, 10). But is this a Pythagorean triad? Let's test it:



6^2 + 8^2

= 36 + 64

= 100

= 10^2



So we know that (6, 8, 10) is also a Pythagorean triad. This works for any value n. You can generate an unlimited number of Pythagorean triads simply by choosing different positive integer values for n, so this is probably sufficient for your purposes.



However, you can't find ALL Pythagorean triads this way.



For instance, consider (7, 24, 25)



7^2 + 24^2

= 49 + 576

= 625

= 25^2



... so it is a Pythagorean triple. But 7 is not a multiple of 3, so we could never find this particular triple using the above method.

Answer 3

This Site Might Help You.



RE:

What are Pythagorean Triads?

I'm doing a maths assignment and one of the questions is to find a method of developing Pythagorean Triads but I dont know what to do! Help!

Answer 4

Triad Definition

Answer 5

http://www.worldofbiography.com/9081-Pythagoras/works.htm

The Pythagorean triads in which the numbers a,b,c do not have a common factor are called primitive Pythagorean triads. For example (3,4,5) (5,12,13) (7,24,25) etc. are primitive Pythagorean triads. But (9,12,15), (15,36,39) are not primitive triads.