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Part D

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 [tex]$\left(x^2-1\right)^2+(2 x)^2=\left(x^2+1\right)^2$[/tex]. Explain your findings.

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] value & Pythagorean Triple \\
\hline
3 & [tex]$(6,8,10)$[/tex] \\
\hline
4 & [tex]$(8,15,17)$[/tex] \\
\hline
5 & [tex]$(10,24,26)$[/tex] \\
\hline
6 & [tex]$(12,35,37)$[/tex] \\
\hline
\end{tabular}

A pattern that I found is that each [tex]$x$[/tex] value multiplied by 2 equals the very first value in the triple.

Sagot :

GinnyAnsweridn
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.