To determine which of the given four sets of side lengths forms a right triangle, we need to verify if any set satisfies the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the two other sides. In mathematical terms, for sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] (where [tex]\(c\)[/tex] is the hypotenuse), the condition [tex]\(a^2 + b^2 = c^2\)[/tex] must hold true.
Let's analyze each set separately:
1. Set 1: [tex]\(6 \, \text{cm}, 7 \, \text{cm}, \sqrt{12} \, \text{cm}\)[/tex]
- The sides are [tex]\(6\)[/tex], [tex]\(7\)[/tex], and [tex]\(\sqrt{12}\)[/tex].
- We need to check if any permutation satisfies [tex]\(a^2 + b^2 = c^2\)[/tex].
2. Set 2: [tex]\(8 \, \text{in}., \sqrt{28} \, \text{in}., \sqrt{35} \, \text{in}.\)[/tex]
- The sides are [tex]\(8\)[/tex], [tex]\(\sqrt{28}\)[/tex], and [tex]\(\sqrt{35}\)[/tex].
- We need to check if any permutation satisfies [tex]\(a^2 + b^2 = c^2\)[/tex].
3. Set 3: [tex]\(\sqrt{3} \, \text{mm}, 4 \, \text{mm}, \sqrt{5} \, \text{mm}\)[/tex]
- The sides are [tex]\(\sqrt{3}\)[/tex], [tex]\(4\)[/tex], and [tex]\(\sqrt{5}\)[/tex].
- We need to check if any permutation satisfies [tex]\(a^2 + b^2 = c^2\)[/tex].
4. Set 4: [tex]\(9 \, \text{ft}, \sqrt{28} \, \text{ft}, 6 \, \text{ft}\)[/tex]
- The sides are [tex]\(9\)[/tex], [tex]\(\sqrt{28}\)[/tex], and [tex]\(6\)[/tex].
- We need to check if any permutation satisfies [tex]\(a^2 + b^2 = c^2\)[/tex].
After performing the necessary calculations for each set, it turns out that none of these sets satisfy the Pythagorean theorem. Hence, none of these sets of side lengths will form a right triangle.
Therefore, the result is:
None of the sets will form a right triangle.
