To determine which of the given triples are Pythagorean triples, we must check if each set of numbers satisfies the Pythagorean theorem. According to the Pythagorean theorem, for a triple [tex]\((a, b, c)\)[/tex] to be a Pythagorean triple, the sum of the squares of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] must equal the square of [tex]\(c\)[/tex]. In other words, [tex]\(a^2 + b^2 = c^2\)[/tex].
Let's examine each triple one by one:
1. Triple [tex]\((8, 15, 17)\)[/tex]:
[tex]\[
8^2 + 15^2 = 64 + 225 = 289
\][/tex]
[tex]\[
17^2 = 289
\][/tex]
Since [tex]\(8^2 + 15^2 = 17^2\)[/tex], [tex]\((8, 15, 17)\)[/tex] is a Pythagorean triple.
2. Triple [tex]\((1, \sqrt{3}, 2)\)[/tex]:
[tex]\[
1^2 + (\sqrt{3})^2 = 1 + 3 = 4
\][/tex]
[tex]\[
2^2 = 4
\][/tex]
Since [tex]\(1^2 + (\sqrt{3})^2 = 2^2\)[/tex], [tex]\((1, \sqrt{3}, 2)\)[/tex] is not a Pythagorean triple.
3. Triple [tex]\((9, 12, 16)\)[/tex]:
[tex]\[
9^2 + 12^2 = 81 + 144 = 225
\][/tex]
[tex]\[
16^2 = 256
\][/tex]
Since [tex]\(9^2 + 12^2\)[/tex] does not equal [tex]\(16^2\)[/tex], [tex]\((9, 12, 16)\)[/tex] is not a Pythagorean triple.
4. Triple [tex]\((8, 11, 14)\)[/tex]:
[tex]\[
8^2 + 11^2 = 64 + 121 = 185
\][/tex]
[tex]\[
14^2 = 196
\][/tex]
Since [tex]\(8^2 + 11^2\)[/tex] does not equal [tex]\(14^2\)[/tex], [tex]\((8, 11, 14)\)[/tex] is not a Pythagorean triple.
5. Triple [tex]\((20, 21, 29)\)[/tex]:
[tex]\[
20^2 + 21^2 = 400 + 441 = 841
\][/tex]
[tex]\[
29^2 = 841
\][/tex]
Since [tex]\(20^2 + 21^2 = 29^2\)[/tex], [tex]\((20, 21, 29)\)[/tex] is a Pythagorean triple.
6. Triple [tex]\((30, 40, 50)\)[/tex]:
[tex]\[
30^2 + 40^2 = 900 + 1600 = 2500
\][/tex]
[tex]\[
50^2 = 2500
\][/tex]
Since [tex]\(30^2 + 40^2 = 50^2\)[/tex], [tex]\((30, 40, 50)\)[/tex] is a Pythagorean triple.
Thus, the triples that are Pythagorean triples are:
- [tex]\((8, 15, 17)\)[/tex]
- [tex]\((20, 21, 29)\)[/tex]
- [tex]\((30, 40, 50)\)[/tex]
