To determine which of the given triples are Pythagorean triples, we need to verify if each triplet [tex]\((a, b, c)\)[/tex] satisfies the Pythagorean theorem, which states that [tex]\(a^2 + b^2 = c^2\)[/tex]. Here is a detailed examination for each given triplet:
1. Triplet [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 triplet.
[tex]\[
\textbf{True}
\][/tex]
2. Triplet [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 a Pythagorean triplet.
[tex]\[
\textbf{False}
\][/tex]
3. Triplet [tex]\((9, 12, 16)\)[/tex]
[tex]\[
9^2 + 12^2 = 81 + 144 = 225
\][/tex]
[tex]\[
16^2 = 256
\][/tex]
Since [tex]\(9^2 + 12^2 \neq 16^2\)[/tex], [tex]\((9, 12, 16)\)[/tex] is not a Pythagorean triplet.
[tex]\[
\textbf{False}
\][/tex]
4. Triplet [tex]\((8, 11, 14)\)[/tex]
[tex]\[
8^2 + 11^2 = 64 + 121 = 185
\][/tex]
[tex]\[
14^2 = 196
\][/tex]
Since [tex]\(8^2 + 11^2 \neq 14^2\)[/tex], [tex]\((8, 11, 14)\)[/tex] is not a Pythagorean triplet.
[tex]\[
\textbf{False}
\][/tex]
5. Triplet [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 triplet.
[tex]\[
\textbf{True}
\][/tex]
6. Triplet [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 triplet.
[tex]\[
\textbf{True}
\][/tex]
Hence, based on the results:
- [tex]\((8, 15, 17)\)[/tex] is a Pythagorean triplet.
- [tex]\((20, 21, 29)\)[/tex] is a Pythagorean triplet.
- [tex]\((30, 40, 50)\)[/tex] is a Pythagorean triplet.
The correct triples are:
[tex]\[
(8, 15, 17), (20, 21, 29), (30, 40, 50)
\][/tex]
