To determine which set of sides fits a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, recall the properties of this specific type of triangle:
- The side opposite the [tex]\(30^\circ\)[/tex] angle (short leg) is half the length of the hypotenuse.
- The side opposite the [tex]\(60^\circ\)[/tex] angle (long leg) is [tex]\(\sqrt{3}\)[/tex] times the short leg.
- The hypotenuse is twice the short leg.
We will evaluate each option to see if they satisfy these properties.
Option A: [tex]\(\frac{1}{2}, \frac{\sqrt{3}}{2}, \sqrt{2}\)[/tex]
For these sides to belong to a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle:
- Short leg: [tex]\(\frac{1}{2}\)[/tex]
- Long leg: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- Hypotenuse should be [tex]\(2 \times \frac{1}{2} = 1\)[/tex], but given is [tex]\(\sqrt{2}\)[/tex]
Since [tex]\(\sqrt{2} \neq 1\)[/tex], this option does not represent the sides of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle.
Option B: [tex]\(3 \sqrt{2}, 3 \sqrt{2}, 6\)[/tex]
For these sides to match the properties of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle:
- If both legs are [tex]\(3\sqrt{2}\)[/tex], this makes it an isosceles triangle, not a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle.
- Therefore, this does not fit the properties.
Option C: [tex]\(\sqrt{3}, 3, 2\sqrt{3}\)[/tex]
For these sides to be correct:
- Assume [tex]\(\sqrt{3}\)[/tex] is the short leg.
- Hypotenuse should be twice the short leg: [tex]\(2 \times \sqrt{3} = 2\sqrt{3}\)[/tex]
- The long leg should be [tex]\(\sqrt{3}\)[/tex] times the short leg: [tex]\(\sqrt{3} \times \sqrt{3} = 3\)[/tex]
This option fits the properties: the short leg is [tex]\(\sqrt{3}\)[/tex], the long leg is [tex]\(3\)[/tex], and the hypotenuse is [tex]\(2\sqrt{3}\)[/tex].
Option D: [tex]\(3, 4, 5\)[/tex]
This represents a Pythagorean triplet and describes a right triangle but not a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle since:
- The hypotenuse should be [tex]\(2 \times 3 = 6\)[/tex], and [tex]\(5 \neq 6\)[/tex]
Therefore, the correct answer is:
C. [tex]\(\sqrt{3}, 3, 2\sqrt{3}\)[/tex]
