To determine which of the given triples are Pythagorean triples, let's recall the definition: A Pythagorean triple consists of three positive integers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] such that [tex]\(a^2 + b^2 = c^2\)[/tex].
Let’s analyze each of the given triples:
1. Triple [tex]\((8, 15, 17)\)[/tex]:
[tex]\[
8^2 + 15^2 = 64 + 225 = 289
\][/tex]
[tex]\[
17^2 = 289
\][/tex]
[tex]\((8, 15, 17)\)[/tex] is a Pythagorean triple because [tex]\(8^2 + 15^2 = 17^2\)[/tex].
2. Triple [tex]\((1, \sqrt{3}, 2)\)[/tex]:
[tex]\[
1^2 + (\sqrt{3})^2 = 1 + 3 = 4
\][/tex]
[tex]\[
2^2 = 4
\][/tex]
[tex]\((1, \sqrt{3}, 2)\)[/tex] is a Pythagorean triple because [tex]\(1^2 + (\sqrt{3})^2 = 2^2\)[/tex].
3. Triple [tex]\((9, 12, 16)\)[/tex]:
[tex]\[
9^2 + 12^2 = 81 + 144 = 225
\][/tex]
[tex]\[
16^2 = 256
\][/tex]
[tex]\((9, 12, 16)\)[/tex] is not a Pythagorean triple because [tex]\(9^2 + 12^2 \neq 16^2\)[/tex].
4. Triple [tex]\((8, 11, 14)\)[/tex]:
[tex]\[
8^2 + 11^2 = 64 + 121 = 185
\][/tex]
[tex]\[
14^2 = 196
\][/tex]
[tex]\((8, 11, 14)\)[/tex] is not a Pythagorean triple because [tex]\(8^2 + 11^2 \neq 14^2\)[/tex].
5. Triple [tex]\((20, 21, 29)\)[/tex]:
[tex]\[
20^2 + 21^2 = 400 + 441 = 841
\][/tex]
[tex]\[
29^2 = 841
\][/tex]
[tex]\((20, 21, 29)\)[/tex] is a Pythagorean triple because [tex]\(20^2 + 21^2 = 29^2\)[/tex].
6. Triple [tex]\((30, 40, 50)\)[/tex]:
[tex]\[
30^2 + 40^2 = 900 + 1600 = 2500
\][/tex]
[tex]\[
50^2 = 2500
\][/tex]
[tex]\((30, 40, 50)\)[/tex] is a Pythagorean triple because [tex]\(30^2 + 40^2 = 50^2\)[/tex].
So the correct Pythagorean triples are:
[tex]\[
(8, 15, 17), \quad (20, 21, 29), \quad (30, 40, 50)
\][/tex]
