To determine which definition best describes Pythagorean triples, let's carefully analyze each option provided:
A. Sets of three whole numbers [tex]$(a, b$[/tex], and [tex]$c$[/tex] ) that satisfy the equation [tex]$\sqrt{a^2+b^2}$[/tex]
This option states that the numbers satisfy the equation [tex]\(\sqrt{a^2 + b^2}\)[/tex]. However, this is not a complete equation and doesn't describe a relationship between [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
B. Any three numbers, each of which is squared
This option simply describes any three numbers that are squared individually but does not provide any relationship among them.
C. Pairs of numbers, [tex]$a$[/tex] and [tex]$b$[/tex], such that [tex]$a^2 = b^2$[/tex]
This option describes pairs of numbers where their squares are equal. This is not relevant to Pythagorean triples as it does not involve three numbers or the specific relationship in the Pythagorean theorem.
D. Sets of three whole numbers ( [tex]$a, b$[/tex], and [tex]$c$[/tex] ) that satisfy the equation [tex]$a^2 + b^2 = c^2$[/tex]
This option correctly describes Pythagorean triples as sets of three whole numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] that satisfy the equation [tex]\(a^2 + b^2 = c^2\)[/tex]. This relationship is the fundamental property of Pythagorean triples, commonly known from the Pythagorean theorem in geometry.
Therefore, the correct answer is:
D. Sets of three whole numbers ( [tex]$a, b$[/tex], and [tex]$c$[/tex] ) that satisfy the equation [tex]\(a^2 + b^2 = c^2\)[/tex].
