what is a pythagorean triplet?

Answer 1

A Pythagorean triplet is a set of three whole numbers, say A, B and C, such that A^2 + B^2 = C^2. They form the sides of a right-angled triangle.

Here is how to construct an infinite number of such triplets. Take any two numbers P and Q, one of them even and the other one odd. Calculate the numbers A = P^2 - Q^2, B = 2 * P * Q, and C = P^2 + Q^2. You will find that in every possible case, A^2 + B^2 = C^2.

If P and Q have no common factor, then neither will A, B and C, in which case they are called a PRIMITIVE Pythagorean triplet.

Example 1: P = 6, Q = 11, A = 85, B = 132, C = 157, no common factor.
Example 2: P = 6, Q = 15, A = 189, B = 180, C = 261, common factor 9.

Answer 2

For the best answers, search on this site https://shorturl.im/aybbp

There is a whole sequence of them, which you can generate yourself (i.e. you don't need to remember anything) in which the longer leg and the hypoteneuse differ by 1 only (as these two do). Notice that 3^2 = 9 = 4 + 5 and that 5^2 = 25 = 12 + 13 So the simple generating formula is to take any odd number, square it, halve it, and take the number either side. The next one is 7, 24, 25 7^2 = 49 49 = 24 + 25 The next three are 9, 40, 41 11, 60, 61, 13, 84, 84 You will notice that there is an arithmetic progression involved in successive triplets: 3, 4, 5 to 5, 12, 13 increments by 8 5, 12, 13 to 7, 24, 25 increments by 12 7, 24, 25 to 9, 40, 41 increments by 16 9, 40, 41 to 11, 60, 61 increments by 20 11, 60, 61 to 13, 84, 85 increments by 24 so we can predict that the next triplet will increment by 28 and be 15, 112, 113 and indeed 15^2 = 225 = 112 + 113 There is also a sequence of triples where the longer leg and the hypoteneuse differ by 2 and the shorter leg is even. The rule this time is "take any even number divisible by 4, halve it, square it, take the number either side" 4 generates 2, 2^2 = 4 and 3 and 5 as the other two sides with a triplet (4, 3, 5)(surprise surprise). 8 generates 4, 4^2 = 16, and 15 and 17 as the other two sides with a triplet (8, 15, 17). 12 generates 6, 6^2 = 36 and 35 and 37 as the other two sides with a triplet (12, 35, 37). And the next three are (16, 63, 65) (20, 99, 101) and (24, 143, 145) all of which you can do in your head, as most people know up to their twelve times table. NB the incremental steps go up by 8: 4, 3, 5 to 8, 15, 17 is a step of 12 8, 15, 17 to 12, 35, 37 is a step of 20 12, 35, 37 to 16, 63, 65 is a step of 28 16, 63, 65 to 20, 99, 101 is a step of 36 20, 99, 101 to 24, 143, 145 is a step of 44 NB2 you can generate triples starting with short legs of even numbers not divisible by 4: (6, 8, 10) (10, 24, 26) (14, 48, 50) but these not Primitive Triples, they are simply double (3, 4, 5) double (5, 12, 13) and double (7, 24, 25) respectively and as such are not particularly interesting. A third sequence involves the two legs differing by 1 only and the first of these is again (3, 4, 5) and the next is (20, 21, 29) (put these numbers into the Theorem: 400 + 441 = 841) The one after that is (119, 120, 169). As you will notice they get larger, rapidly.The generating formula, though simple, involves a little arithmetic. For a triplet (a, b. c) with this property to be preserved in the next triplet (A, B, C) generated A = 2a + b + 2c B = a + 2b + 2c C = 2a + 2b + 3c with (a, b, c) = 3, 4, 5 A = 6 + 4 + 10 = 20 B = 3 + 8 + 10 = 21 C = 6 + 8 + 15 = 29 with (a, b, c) = 20, 21, 29 A = 40 + 21 + 58 = 119 B = 20 + 42 + 58 = 120 C = 40 + 42 + 87 = 169 See if you can generate the next two or three in this sequence yourself! TO GENERALISE Similar formulae exist to generate all possible triples from (3, 4, 5) e.g. the ones Cassie has listed below (the 16 triples with all sides under 100) . Every triple can generate three others. And each of those can generate three more. This is what is called a trinary tree. To generate (D, E, F) from (a, b, c) and preserve the higher interval (hypoteneuse - longer leg) D = 2a - b + 2c E = a - 2b + 2c F = 2a - 2b + 3c eg if (a b c) = 5, 12, 13) D = 10 - 12 + 26 = 24 E = 5 - 24 + 26 = 7 F = 10 - 24 + 39 = 25 with a triple of (7, 24, 25) preserving the higher interval of 1 from, (5, 12, 13). To generate (G, H, I) from (a, b, c) and preserve the widest interval (hypoteneuse - shorter leg) (as the higher interval of the new triple): G = - 2a + b + 2c H = - a + 2b + 2c I = - 2a + 2b + 3c With (a b c) = (5, 12, 13) G = - 10 + 12 + 26 = 28 H = -5 + 24 + 26 = 45 I = - 10 + 24 + 39 = 53 (with a triple (28, 45, 53) with the 8 interval from (5, 12, 13) preserved And the third triple that can be generated preserves the lower interval (Longer leg - shorter leg) which we have already met when that difference was 1 and was preserved. Trying that again with the lower interval of 7 in (5, 12, 13) as (a b c) we find: A = 2a + b + 2c = 10 + 12 + 26 = 48 B = a + 2b + 2c = 5 + 24 + 26 = 55 C = 2a + 2b + 3c = 10 + 24 + 39 = 73 with a triple of (48, 55, 73) and the lower interval of 7 preserved as it is in (5, 12, 13) All Cassie's other triples can be generated using these three sets of equations for (A, B, C) (D, E, F) and (G, H, I) and indeed all triples can thus be generated from the descendants of (3, 4, 5) viz. (5, 12, 13) (8, 15, 17) and (20, 21, 29). Kinda neat, huh?

Answer 3

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime.

The name is derived from the Pythagorean theorem, of which every Pythagorean triple is a solution. The converse is not true. For instance, the triangle with sides a = b = 1 and c = √2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. Moreover, 1 and √2 don't have an integer common multiple because √2 is irrational. There are 16 primitive Pythagorean triples with c ≤ 100:

(3, 4, 5) (20, 21, 29) (11, 60, 61) (13, 84, 85)
(5, 12, 13) (12, 35, 37) (16, 63, 65) (36, 77, 85)
(8, 15, 17) (9, 40, 41) (33, 56, 65) (39, 80, 89)
(7, 24, 25) (28, 45, 53) (48, 55, 73) (65, 72, 97)

Generating Pythagorean triples

It is primitive if and only if m and n are relatively prime and one of them is even (if both n and m are odd, then a, b, and c will be even, and so the Pythagorean triple will not be primitive). Not every Pythagorean triple can be generated in this way, but every primitive triple (possibly after exchanging a and b) arises in this fashion from a unique pair of coprime numbers m > n. This shows that there are infinitely many primitive Pythagorean triples. This formula was given by Euclid (c. 300 B.C.) in his book Elements and is referred to as Euclid's formula.

Also, it is easy to notice that the complex number m + in when squared gives a + ib as a result. Since | z2 | = | z | 2, is an integer.

Properties of Pythagorean triples-

The properties of primitive Pythagorean triples include:

-Exactly one of a, b is odd; c is odd.
-The area (A = ab/2) is an integer.
-Exactly one of a, b is divisible by 3.
-Exactly one of a, b is divisible by 4.
-Exactly one of a, b, c is divisible by 5.
-For any Pythagorean triple, ab is divisible by 12, and abc is divisible by 60.
-Exactly one of a, b, (a + b), (a − b) is divisible by 7.
-At most one of a, b is a square.
-Every integer greater than 2 is part of a Pythagorean triple.
-There exist infinitely many Pythagorean triples whose hypotenuses are squares of natural numbers.
-There exist infinitely many Pythagorean triples in which one of the legs is the square of a natural number.
-For each natural number n, there exist n Pythagorean triples with different hypotenuses and the same area.
-For each natural number n, there exist at least n different Pythagorean triples with the same leg a, where a is some natural number
-For each natural number n, there exist at least n different triangles with the same hypotenuse.
-In every Pythagorean triple, the radius of the incircle and the radii of the three excircles are natural numbers.
-There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.
-There is no Pythagorean triple in which the hypotenuse is equal to either 2a or 2b..

Answer 4

A Pythagorean triplet is three numbers a,b,andc that satisfy the equation a^2 +b^2 = c^2.

The most common triplet is a=3, b=4,c=5 because 25=16+9.
Another is a=5 b=12 c=13 since 169 = 25 +144.

Multiples of a triple are also a triplet
Since 3,4,5 is a triplet sois 6,8,10 and 9,12,15, etc.

There are formulas for generating triplets but you should commit the most comon ones to memory so that you can quickly solve problem involving right triangl;es.

Answer 5

Pythagorean Triple Definition

Answer 6

This is what I do: 3² = 4 + 5 5² = 12 + 13 7² = 24 + 25 9² = 40 + 41 Notice the pattern? The trick to notice is that the smallest value should be an odd number. Then divide the square of that 'almost evenly' so that we can get the next two numbers. The last two are consecutive. Now, I had given you the trick to remember infinitely many triples. (I think it is called Pythagorean triples, but whatever...) Maybe you could try it: {11, __ , __ } The only triple that I know that does not follow that rule is {8,15,17} and I remember that because it is the only simplified triple that starts with an even number. Edit: Hey, Cassie gave you even more triples that start with an even number. But you only need the small ones anyway.

Answer 7

"Pythagorean triplets" are integer solutions to the Pythagorean Theorem,
a2 + b2 = c2

e.g
3 4 5
5 12 13
7 24 25
9 40 41
11 60 61


Greetings from Germany
Tobi

Answer 8

I liked bh8153's answer. Not only did you get your answer, but you can construct as many triples as you want.

You can expand that to quadruples, as well. Just as the Pythagorean theorem can be used to find the diagonal of a square, it can also be used to find the diagonal of a cube (If you step thruough the process, of finding the diagonal, you'll see why this has to be true). In this case, the quadruple would be:

Where m,n, and p are any 3 integers:

a=2mp
b=2np
c=p^2 - (m^2 + n^2)
d=p^2 + m^2 +n^2

For example, if m= 1, n=2, and p=3, then a cube with sides of 6, 12, and 4 would have a diagonal of 14:

6^2 + 12^2 + 4^2 = 14^2

Answer 9

The brother or sister of a pythagorean twin. : )

A Pythagorean triplet is three numbers a,b,andc that satisfy the equation a^2 +b^2 = c^2.

Answer 10

What Is Pythagorean Triple