To determine which set of numbers could not represent the three sides of a right triangle, we need to apply the Pythagorean theorem. The Pythagorean theorem states that for a right triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and hypotenuse [tex]\(c\)[/tex]:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
We will verify this equation for each of the given sets of numbers:
1. Set: \{57, 76, 95\}
Let's check if [tex]\(57^2 + 76^2 = 95^2\)[/tex]:
[tex]\[ 57^2 = 3249 \][/tex]
[tex]\[ 76^2 = 5776 \][/tex]
[tex]\[ 95^2 = 9025 \][/tex]
[tex]\[ 3249 + 5776 = 9025 \][/tex]
Since [tex]\(3249 + 5776 = 9025\)[/tex], this set satisfies the Pythagorean theorem and can represent the sides of a right triangle.
2. Set: \{10, 25, 26\}
Let's check if [tex]\(10^2 + 25^2 = 26^2\)[/tex]:
[tex]\[ 10^2 = 100 \][/tex]
[tex]\[ 25^2 = 625 \][/tex]
[tex]\[ 26^2 = 676 \][/tex]
[tex]\[ 100 + 625 = 725 \][/tex]
Since [tex]\(100 + 625 \neq 676\)[/tex], this set does not satisfy the Pythagorean theorem and cannot represent the sides of a right triangle.
3. Set: \{18, 80, 82\}
Let's check if [tex]\(18^2 + 80^2 = 82^2\)[/tex]:
[tex]\[ 18^2 = 324 \][/tex]
[tex]\[ 80^2 = 6400 \][/tex]
[tex]\[ 82^2 = 6724 \][/tex]
[tex]\[ 324 + 6400 = 6724 \][/tex]
Since [tex]\(324 + 6400 = 6724\)[/tex], this set satisfies the Pythagorean theorem and can represent the sides of a right triangle.
4. Set: \{65, 72, 97\}
Let's check if [tex]\(65^2 + 72^2 = 97^2\)[/tex]:
[tex]\[ 65^2 = 4225 \][/tex]
[tex]\[ 72^2 = 5184 \][/tex]
[tex]\[ 97^2 = 9409 \][/tex]
[tex]\[ 4225 + 5184 = 9409 \][/tex]
Since [tex]\(4225 + 5184 = 9409\)[/tex], this set satisfies the Pythagorean theorem and can represent the sides of a right triangle.
Therefore, the set \{10, 25, 26\} cannot represent the three sides of a right triangle.
