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Is there a formula to find all the triples?
Take any two numbers, for example, n and m, and n>m and then find:
- 2mn
- n² − m²
- n² +m²
For example: Take n=2 m =1 2(1*2) = 4 2 squared = 4 - 1 squared = 4-1 =3 and 2 squared + 1 squared =5 so you have 3,4,5
List of Common Pythagorean Triples
|
Triple |
Triple x2 |
Triple x 3 |
Triple x 4 |
|
3, 4, 5 |
6,8,10 |
9,12,15 |
12,16,20 |
|
5,12,13 |
10,24,26 |
15,36,39 |
20,48,52 |
|
7,24,25 |
14,48,50 |
21,72,75 |
28,96,100 |
|
8,15,17 |
16,30,34 |
24,45,51 |
32,60,68 |
|
9,40,41 |
18,80,82 |
27,120,123 |
36,160,164 |
|
11,60,61 |
22,120,122 |
33,180,183 |
44,240,244 |
|
12,35,37 |
24,70,74 |
36,105,111 |
48,140,148 |
|
13,84,85 |
26,168,170 |
39,252,255 |
52,336,340 |
|
16,63,65 |
32,126,130 |
48,189,195 |
64,252,260 |
|
20,21,29 |
40,42,58 |
60,63,87 |
80,84,116 |
Pythagorean Triples are positive integers that satisfy the Pythagorean Theorem, and any multiples of these numbers also fulfill the Pythagorean Theorem.
Take the numbers 3, 4, and 5
3 squared=9 4 squared=16 and 5 squared =25 so 9+16=25 and therefore this triplet of numbers satisfies the Pythagorean Theorem
If you multiply all three numbers by 3 (9, 12, and 15), these new numbers also fulfill the Pythagorean Theorem. 9 squared = 81 12 squared = 144 15 squared =225
81 + 144 =225
Transcript Common Triples
Hi welcome to MooMooMath. Today we are going to look at common triples which are associated with the Pythagorean Theorem. Here is a common triple., a three four five which works in the Pythagorean theorem because 3 squared ( 9 ) plus four squared ( 16 ) equals 5 squared ( 25 16 + 9 equals 25 ) so we have the numbers three. Four and five which are a common triple. In the Pythagorean Theorem. Below it are a triangle with a side of 6 and a hypotenuse of 10 and an X as the unknown side of X if you will notice this shows you a common triple. So three and 6 are associated with each other so they are corresponding sides and if I double 3 I get 6 and if I double 5 I get 10. Therefore if I double 4 I get 8 so the missing side is 8 so this is just applying the Pythagorean Theorem triple to an actual problem. So what are the actual rules for doing this? So what you will do is take your common triplets and multiple each number by the same factor. So we have a three, four, and five and in the example we multiplied each side by two to get a six, eight, ten triangle. You can also go back and multiple 3, 4, 5 by three and get 9, 12. and 15. You get do that with 4 10 or a 40, 50 right triangle. So what are the common triples? Let me show you several of the common triples you will see. 3, 4, 5 and multiples of those. A 5, 12, 13 is also a common triple. Because 5 squared plus 12 squared equals 13 squared. 25 plus 144 equals 169. Here are three more common triples 7, 24, 25 the 8, 15, 17 and the 20, 21, 29. So those are common triples you can take and multiple by sides by common factors. So you can see how this is done. Since I showed you the 3, 4, 5 triple first this time I will use the 5, 12, and 13. So if I were to make a table of possible values I will just draw the 5, 12, and 13 on top and make a list. If I multiple by two I get 10, 24, and 26. And multiple by three I get 15 26, 39 and by 4 I get 20 48 and 52 and those would be our common triples of 5, 12, 13. Hope this was helpful
Pythagorean Triples List
Why use Pythagorean triples?
Speed…..if you recognize a common triple, it saves you time working out the Pythagorean Theorem.
Number Sense…knowing Pythagorean triples is a form of number sense.
When solving a right triangle, look for common factors in the given sides….
Ex. Given a leg of 9 and hypotenuse of 15, both are divisible by 3.
9/3 = 3 and 15/3 = 5, so these two sides are a multiple of the root 3, 4, 5 triangle.
So, multiply the third value, 4 x 3, by the same factor to find the missing side, 12.
What is the formula for Pythagorean Theorem?
a^2 + b^2 = c^2
a
b
c
The square of leg a plus the square of leg b equals the square of the hypotenuse c
16
9
25
Pythagorean Theorem Visually
Pythagorean Theorem Formula