Question

Examine this set of Pythagorean triples. Look for a pattern that is true for each triple regarding the difference between the three values that make up the triple.
Describe this pattern. Then see if you can think of another Pythagorean triple that doesn’t follow the pattern you just described and that can’t be generated using the identity (x2 − 1)2 + (2x)2 = (x2 + 1)2. Explain your findings.

I have attached an image of the triples. If anyone could help me with this, I'd greatly appreciate it. Please respond correctly. Tysm.

Answer 1

A Pythagorean triplet is a set of 3 positive integer numbers which may be the sides of a right triangle, i.e. they meet the Pythagorean theorem c² = a² + b².

You can check that the numbers on your table are Pythagorean triplets by substituting them in the Pythagorean equation:

1. 6² + 8² = 36 + 64 = 100 = 10²
2. 8² + 15² = 64 + 225 = 289 = 17²
3. 10² + 24² = 100 + 576 = 676 = 26²
4. 12² + 35² = 144 + 1225 = 1369 = 37²

Now, lets look for the pattern:

x-value     Pythagorean

                 triple    

3               (6, 8, 10)              6/2 = 3

                                            3² - 1 = 9 - 1 = 8

                                            3² + 1 = 9 + 1 = 10

----------------------------------------------------------------------

4               (8, 15, 17)              8/2 = 4

                                             4² - 1 = 16 - 1 = 15

                                             4² + 1 = 16 + 1 = 17

---------------------------------------------------------------------

5               (10, 24, 26)              10/2 = 5

                                                 5² - 1 = 25 - 1 = 24

                                                 24² + 1   = 25 + 1 = 26

--------------------------------------------------------------------------

6               (12, 35, 37)              12/2 = 6

                                                6² - 1 = 36 - 1 = 35

                                               6² + 1   = 36 + 1 = 37

----------------------------------------------------------------------

From which you find the pattern: the first number is 2x, the second number is x² - 1, and the third number is x² + 1

                                            ⇒ (2x)² + (x² - 1)² = (x² + 1)², or

                                                (x² - 1)² + (2x)² = (x² + 1)².

Other example of a Pythagorean triple is (3, 4, 5). You migth think that it does not follow the pattern, but if you do x = 2, you end with:

• x = 2
• 2x =  2(2) = 4
• x² - 1 = 2² - 1 = 3
• x² + 1 = 2² + 1 = 5

Hence, (3, 4, 5) also follows the pattern.

Only  right triangles with non-integer sides do not form Pythagorean triples.

Of course you may proof that (x² - 1)² + (2x)² = (x² + 1)² is an identity (always true):

Left hand side: (x⁴ - 2x² + 1) + 4x² = x⁴ + 2x² + 1

Right hand side: x⁴ + 2x² + 1

∴ The equation is always true.

At the end, the pattern is true for any Pythagorean triplet, but a more formal proof is beyond the scope of this question.