Find the best solutions to your problems with the help of IDNLearn.com's expert users. Our platform offers detailed and accurate responses from experts, helping you navigate any topic with confidence.

Which side lengths form a right triangle? Choose all answers that apply:

A. [tex]5, 6, \sqrt{30}[/tex]
B. [tex]2.5, \sqrt{18}, 5[/tex]
C. [tex]\sqrt{2}, 2, \sqrt{6}[/tex]


Sagot :

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]