To find the Pythagorean triple generated by using [tex]\(x = 4\)[/tex] and [tex]\(y = 1\)[/tex], we use the following formulas:
[tex]\[ a = x^2 - y^2 \][/tex]
[tex]\[ b = 2xy \][/tex]
[tex]\[ c = x^2 + y^2 \][/tex]
Let's go through each step in detail:
1. Calculate [tex]\(a\)[/tex]:
[tex]\[ a = x^2 - y^2 \][/tex]
[tex]\[ a = 4^2 - 1^2 \][/tex]
[tex]\[ a = 16 - 1 \][/tex]
[tex]\[ a = 15 \][/tex]
2. Calculate [tex]\(b\)[/tex]:
[tex]\[ b = 2xy \][/tex]
[tex]\[ b = 2 \cdot 4 \cdot 1 \][/tex]
[tex]\[ b = 8 \][/tex]
3. Calculate [tex]\(c\)[/tex]:
[tex]\[ c = x^2 + y^2 \][/tex]
[tex]\[ c = 4^2 + 1^2 \][/tex]
[tex]\[ c = 16 + 1 \][/tex]
[tex]\[ c = 17 \][/tex]
Therefore, the Pythagorean triple generated by using [tex]\(4\)[/tex] for [tex]\(x\)[/tex] and [tex]\(1\)[/tex] for [tex]\(y\)[/tex] is [tex]\((15, 8, 17)\)[/tex].
From the given choices:
- 3, 5, and 8
- 5, 11, and 12
- 8, 15, and 17
- 6, 8, and 10
The correct Pythagorean triple is:
[tex]\[ \boxed{8, 15, and 17} \][/tex]
