Answer :
Alright, let's analyze the provided Pythagorean triples to identify any patterns and see if there's a way to find a triple that doesn't conform to these patterns.
### Part D: Analysis of Given Pythagorean Triples
The given Pythagorean triples are:
1. [tex]\((6, 8, 10)\)[/tex]
2. [tex]\((8, 15, 17)\)[/tex]
3. [tex]\((10, 24, 26)\)[/tex]
4. [tex]\((12, 35, 37)\)[/tex]
To uncover patterns, we'll first list the differences between the values in each triple to see if any pattern emerges.
#### Differences
1. [tex]\((6, 8, 10)\)[/tex]
- [tex]\(8 - 6 = 2\)[/tex]
- [tex]\(10 - 8 = 2\)[/tex]
2. [tex]\((8, 15, 17)\)[/tex]
- [tex]\(15 - 8 = 7\)[/tex]
- [tex]\(17 - 15 = 2\)[/tex]
3. [tex]\((10, 24, 26)\)[/tex]
- [tex]\(24 - 10 = 14\)[/tex]
- [tex]\(26 - 24 = 2\)[/tex]
4. [tex]\((12, 35, 37)\)[/tex]
- [tex]\(35 - 12 = 23\)[/tex]
- [tex]\(37 - 35 = 2\)[/tex]
From this, we can see that in each triple, the difference between the last two values (i.e., the longer leg and the hypotenuse) is always [tex]\(2\)[/tex]. This gives us a pattern:
Pattern Identified: The difference between the second and third values of each Pythagorean triple is consistently [tex]\(2\)[/tex].
### Part E: Counterexample Pythagorean Triple
Next, let's identify a Pythagorean triple that does not follow this pattern and is not generated using the identity [tex]\((x^2 - 1)^2 + (2x)^2 = (x^2 + 1)^2\)[/tex].
Consider the Pythagorean triple [tex]\((5, 12, 13)\)[/tex].
1. Check the differences:
- [tex]\(12 - 5 = 7\)[/tex]
- [tex]\(13 - 12 = 1\)[/tex]
Clearly:
- [tex]\(13 - 12 = 1\)[/tex], which is not [tex]\(2\)[/tex].
### Conclusion
We found that the difference between the second and third values (i.e., the longer leg and the hypotenuse) in each provided Pythagorean triple is always [tex]\(2\)[/tex]. However, the Pythagorean triple [tex]\((5, 12, 13)\)[/tex] does not conform to this pattern; here, the difference between the longer leg and the hypotenuse is [tex]\(1\)[/tex]. Additionally, [tex]\((5, 12, 13)\)[/tex] cannot be derived using the identity [tex]\((x^2 - 1)^2 + (2x)^2 = (x^2 + 1)^2\)[/tex].
This demonstrates that while certain Pythagorean triples may follow a discernible pattern, not all Pythagorean triples adhere to the same rules. There are indeed exceptions, such as [tex]\((5, 12, 13)\)[/tex], which showcases the diversity and complexity within Pythagorean triples.
### Part D: Analysis of Given Pythagorean Triples
The given Pythagorean triples are:
1. [tex]\((6, 8, 10)\)[/tex]
2. [tex]\((8, 15, 17)\)[/tex]
3. [tex]\((10, 24, 26)\)[/tex]
4. [tex]\((12, 35, 37)\)[/tex]
To uncover patterns, we'll first list the differences between the values in each triple to see if any pattern emerges.
#### Differences
1. [tex]\((6, 8, 10)\)[/tex]
- [tex]\(8 - 6 = 2\)[/tex]
- [tex]\(10 - 8 = 2\)[/tex]
2. [tex]\((8, 15, 17)\)[/tex]
- [tex]\(15 - 8 = 7\)[/tex]
- [tex]\(17 - 15 = 2\)[/tex]
3. [tex]\((10, 24, 26)\)[/tex]
- [tex]\(24 - 10 = 14\)[/tex]
- [tex]\(26 - 24 = 2\)[/tex]
4. [tex]\((12, 35, 37)\)[/tex]
- [tex]\(35 - 12 = 23\)[/tex]
- [tex]\(37 - 35 = 2\)[/tex]
From this, we can see that in each triple, the difference between the last two values (i.e., the longer leg and the hypotenuse) is always [tex]\(2\)[/tex]. This gives us a pattern:
Pattern Identified: The difference between the second and third values of each Pythagorean triple is consistently [tex]\(2\)[/tex].
### Part E: Counterexample Pythagorean Triple
Next, let's identify a Pythagorean triple that does not follow this pattern and is not generated using the identity [tex]\((x^2 - 1)^2 + (2x)^2 = (x^2 + 1)^2\)[/tex].
Consider the Pythagorean triple [tex]\((5, 12, 13)\)[/tex].
1. Check the differences:
- [tex]\(12 - 5 = 7\)[/tex]
- [tex]\(13 - 12 = 1\)[/tex]
Clearly:
- [tex]\(13 - 12 = 1\)[/tex], which is not [tex]\(2\)[/tex].
### Conclusion
We found that the difference between the second and third values (i.e., the longer leg and the hypotenuse) in each provided Pythagorean triple is always [tex]\(2\)[/tex]. However, the Pythagorean triple [tex]\((5, 12, 13)\)[/tex] does not conform to this pattern; here, the difference between the longer leg and the hypotenuse is [tex]\(1\)[/tex]. Additionally, [tex]\((5, 12, 13)\)[/tex] cannot be derived using the identity [tex]\((x^2 - 1)^2 + (2x)^2 = (x^2 + 1)^2\)[/tex].
This demonstrates that while certain Pythagorean triples may follow a discernible pattern, not all Pythagorean triples adhere to the same rules. There are indeed exceptions, such as [tex]\((5, 12, 13)\)[/tex], which showcases the diversity and complexity within Pythagorean triples.
