To determine which sets of side lengths form a right triangle, we need to check if they satisfy the Pythagorean theorem. The Pythagorean theorem states that for a triangle to be a right triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], where [tex]\(c\)[/tex] is the hypotenuse (the longest side), the following equation must hold:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Let's evaluate each set of side lengths:
### Set A: [tex]\(5, 6, \sqrt{30}\)[/tex]
1. Identify the largest side: [tex]\(\sqrt{30}\)[/tex] (approximately 5.477).
2. Check the Pythagorean theorem:
[tex]\[ 5^2 + 6^2 = \sqrt{30}^2 \][/tex]
[tex]\[ 25 + 36 = 30 \][/tex]
[tex]\[ 61 \neq 30 \][/tex]
Clearly, [tex]\(5^2 + 6^2 \neq \sqrt{30}^2\)[/tex].
### Set B: [tex]\(2.5, \sqrt{18}, 5\)[/tex]
1. Identify the largest side: [tex]\(5\)[/tex].
2. Check the Pythagorean theorem:
[tex]\[ 2.5^2 + (\sqrt{18})^2 = 5^2 \][/tex]
[tex]\[ 6.25 + 18 = 25 \][/tex]
[tex]\[ 24.25 \neq 25 \][/tex]
Clearly, [tex]\(2.5^2 + (\sqrt{18})^2 \neq 5^2\)[/tex].
### Set C: [tex]\(\sqrt{2}, 2, \sqrt{6}\)[/tex]
1. Identify the largest side: [tex]\(\sqrt{6}\)[/tex] (approximately 2.449).
2. Check the Pythagorean theorem:
[tex]\[ (\sqrt{2})^2 + 2^2 = (\sqrt{6})^2 \][/tex]
[tex]\[ 2 + 4 = 6 \][/tex]
[tex]\[ 6 = 6 \][/tex]
In this case, [tex]\((\sqrt{2})^2 + 2^2 = (\sqrt{6})^2\)[/tex].
Based on our evaluations, only the set [tex]\( \sqrt{2}, 2, \sqrt{6} \)[/tex] satisfies the Pythagorean theorem and thus forms a right triangle.
Therefore, the correct answer is:
c. [tex]\(\sqrt{2}, 2, \sqrt{6}\)[/tex]
