To determine which sets of numbers form Pythagorean triples, we need to verify if each set satisfies the Pythagorean theorem, where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the side lengths of a triangle, with [tex]\(c\)[/tex] being the hypotenuse. The Pythagorean theorem states that:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
We will check each set individually:
### Set 1: [tex]\(51, 45, 24\)[/tex]
Sort the numbers so we assume [tex]\(24\)[/tex] and [tex]\(45\)[/tex] are the legs and [tex]\(51\)[/tex] is the hypotenuse:
[tex]\[ 24^2 + 45^2 = 51^2 \][/tex]
[tex]\[ 576 + 2025 = 2601 \][/tex]
[tex]\[ 2601 = 2601 \][/tex]
This set satisfies the Pythagorean theorem, so [tex]\(51, 45, 24\)[/tex] is a Pythagorean triple.
### Set 2: [tex]\(8 \sqrt{5}, 8, 16\)[/tex]
Sort the numbers so we assume [tex]\(8\sqrt{5}\)[/tex] and [tex]\(8\)[/tex] are the legs and [tex]\(16\)[/tex] is the hypotenuse:
[tex]\[ (8\sqrt{5})^2 + 8^2 = 16^2 \][/tex]
[tex]\[ 320 + 64 = 256 \][/tex]
[tex]\[ 384 \neq 256 \][/tex]
This set does not satisfy the Pythagorean theorem.
### Set 3: [tex]\(4 \sqrt{13}, 8, 12\)[/tex]
Sort the numbers so we assume [tex]\(4\sqrt{13}\)[/tex] and [tex]\(8\)[/tex] are the legs and [tex]\(12\)[/tex] is the hypotenuse:
[tex]\[ (4\sqrt{13})^2 + 8^2 = 12^2 \][/tex]
[tex]\[ 208 + 64 = 144 \][/tex]
[tex]\[ 272 \neq 144 \][/tex]
This set does not satisfy the Pythagorean theorem.
### Set 4: [tex]\(26, 10, 24\)[/tex]
Sort the numbers so we assume [tex]\(10\)[/tex] and [tex]\(24\)[/tex] are the legs and [tex]\(26\)[/tex] is the hypotenuse:
[tex]\[ 10^2 + 24^2 = 26^2 \][/tex]
[tex]\[ 100 + 576 = 676 \][/tex]
[tex]\[ 676 = 676 \][/tex]
This set satisfies the Pythagorean theorem, so [tex]\(26, 10, 24\)[/tex] is a Pythagorean triple.
### Set 5: [tex]\(15, 20, 25\)[/tex]
Sort the numbers so we assume [tex]\(15\)[/tex] and [tex]\(20\)[/tex] are the legs and [tex]\(25\)[/tex] is the hypotenuse:
[tex]\[ 15^2 + 20^2 = 25^2 \][/tex]
[tex]\[ 225 + 400 = 625 \][/tex]
[tex]\[ 625 = 625 \][/tex]
This set satisfies the Pythagorean theorem, so [tex]\(15, 20, 25\)[/tex] is a Pythagorean triple.
### Conclusion
The sets that form Pythagorean triples are:
- [tex]\(51, 45, 24\)[/tex]
- [tex]\(26, 10, 24\)[/tex]
- [tex]\(15, 20, 25\)[/tex]
