To generate a Pythagorean triple using the formulas [tex]\(a = x^2 - y^2\)[/tex], [tex]\(b = 2xy\)[/tex], and [tex]\(c = x^2 + y^2\)[/tex] with [tex]\(x = 4\)[/tex] and [tex]\(y = 1\)[/tex], let's go through each step in detail.
1. Calculate [tex]\(a\)[/tex]:
[tex]\[
a = x^2 - y^2
\][/tex]
Substitute [tex]\(x = 4\)[/tex] and [tex]\(y = 1\)[/tex]:
[tex]\[
a = 4^2 - 1^2 = 16 - 1 = 15
\][/tex]
2. Calculate [tex]\(b\)[/tex]:
[tex]\[
b = 2xy
\][/tex]
Substitute [tex]\(x = 4\)[/tex] and [tex]\(y = 1\)[/tex]:
[tex]\[
b = 2 \cdot 4 \cdot 1 = 2 \cdot 4 = 8
\][/tex]
3. Calculate [tex]\(c\)[/tex]:
[tex]\[
c = x^2 + y^2
\][/tex]
Substitute [tex]\(x = 4\)[/tex] and [tex]\(y = 1\)[/tex]:
[tex]\[
c = 4^2 + 1^2 = 16 + 1 = 17
\][/tex]
Therefore, the Pythagorean triple generated by using [tex]\(x = 4\)[/tex] and [tex]\(y = 1\)[/tex] is [tex]\(a = 15\)[/tex], [tex]\(b = 8\)[/tex], and [tex]\(c = 17\)[/tex].
From the given options, the correct Pythagorean triple is:
[tex]\[ \boxed{8, 15, 17} \][/tex]
