Answer :
To determine which set of values could be the side lengths of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, we must use the properties of such a triangle. In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, the sides are in the ratio of [tex]\(1 : \sqrt{3} : 2\)[/tex]. Specifically:
- The side opposite the [tex]\(30^\circ\)[/tex] angle is the shortest side, denoted as [tex]\(a\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle is [tex]\(\sqrt{3}\)[/tex] times the shortest side, [tex]\(a \sqrt{3}\)[/tex].
- The side opposite the [tex]\(90^\circ\)[/tex] angle (the hypotenuse) is twice the shortest side, [tex]\(2a\)[/tex].
Given this, let's check each option to see if the specified lengths match these properties:
Option A: [tex]\(\{6, 12, 12 \sqrt{2}\}\)[/tex]
- Here, [tex]\(a = 6\)[/tex].
- The longest side should be [tex]\(2 \times 6 = 12\)[/tex], which matches one of the sides.
- The other side should be [tex]\(6 \sqrt{3}\)[/tex], but here it is given as [tex]\(12 \sqrt{2}\)[/tex].
Since [tex]\(12 \sqrt{2} \neq 6 \sqrt{3}\)[/tex], this option is not valid.
Option B: [tex]\(\{6, 6 \sqrt{2}, 12\}\)[/tex]
- Here, [tex]\(a = 6\)[/tex].
- The longest side should be [tex]\(2 \times 6 = 12\)[/tex], which matches one of the sides.
- The other side should be [tex]\(6 \sqrt{3}\)[/tex], but here it is given as [tex]\(6 \sqrt{2}\)[/tex].
Since [tex]\(6 \sqrt{2} \neq 6 \sqrt{3}\)[/tex], this option is not valid.
Option C: [tex]\(\{6, 12, 12 \sqrt{3}\}\)[/tex]
- Here, [tex]\(a = 6\)[/tex].
- The longest side should be [tex]\(2 \times 6 = 12\)[/tex], but here it is given as [tex]\(12 \sqrt{3}\)[/tex].
- The side [tex]\(12\)[/tex] should be [tex]\(6 \sqrt{3}\)[/tex], but the given side [tex]\(12\)[/tex] does not conform to [tex]\(6 \sqrt{3}\)[/tex].
Therefore, this option is not valid either.
Option D: [tex]\(\{6, 6 \sqrt{5}, 12\}\)[/tex]
- Here, [tex]\(a = 6\)[/tex].
- The longest side should be [tex]\(2 \times 6 = 12\)[/tex], which matches one of the sides.
- The other side should be [tex]\(6 \sqrt{3}\)[/tex], but here it is given as [tex]\(6 \sqrt{5}\)[/tex].
Since [tex]\(6 \sqrt{5} \neq 6 \sqrt{3}\)[/tex], this option is not valid.
Given the analysis above, none of the provided sets of values could be the side lengths of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle. Therefore, the answer is:
[tex]\[ \boxed{\text{None}} \][/tex]
- The side opposite the [tex]\(30^\circ\)[/tex] angle is the shortest side, denoted as [tex]\(a\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle is [tex]\(\sqrt{3}\)[/tex] times the shortest side, [tex]\(a \sqrt{3}\)[/tex].
- The side opposite the [tex]\(90^\circ\)[/tex] angle (the hypotenuse) is twice the shortest side, [tex]\(2a\)[/tex].
Given this, let's check each option to see if the specified lengths match these properties:
Option A: [tex]\(\{6, 12, 12 \sqrt{2}\}\)[/tex]
- Here, [tex]\(a = 6\)[/tex].
- The longest side should be [tex]\(2 \times 6 = 12\)[/tex], which matches one of the sides.
- The other side should be [tex]\(6 \sqrt{3}\)[/tex], but here it is given as [tex]\(12 \sqrt{2}\)[/tex].
Since [tex]\(12 \sqrt{2} \neq 6 \sqrt{3}\)[/tex], this option is not valid.
Option B: [tex]\(\{6, 6 \sqrt{2}, 12\}\)[/tex]
- Here, [tex]\(a = 6\)[/tex].
- The longest side should be [tex]\(2 \times 6 = 12\)[/tex], which matches one of the sides.
- The other side should be [tex]\(6 \sqrt{3}\)[/tex], but here it is given as [tex]\(6 \sqrt{2}\)[/tex].
Since [tex]\(6 \sqrt{2} \neq 6 \sqrt{3}\)[/tex], this option is not valid.
Option C: [tex]\(\{6, 12, 12 \sqrt{3}\}\)[/tex]
- Here, [tex]\(a = 6\)[/tex].
- The longest side should be [tex]\(2 \times 6 = 12\)[/tex], but here it is given as [tex]\(12 \sqrt{3}\)[/tex].
- The side [tex]\(12\)[/tex] should be [tex]\(6 \sqrt{3}\)[/tex], but the given side [tex]\(12\)[/tex] does not conform to [tex]\(6 \sqrt{3}\)[/tex].
Therefore, this option is not valid either.
Option D: [tex]\(\{6, 6 \sqrt{5}, 12\}\)[/tex]
- Here, [tex]\(a = 6\)[/tex].
- The longest side should be [tex]\(2 \times 6 = 12\)[/tex], which matches one of the sides.
- The other side should be [tex]\(6 \sqrt{3}\)[/tex], but here it is given as [tex]\(6 \sqrt{5}\)[/tex].
Since [tex]\(6 \sqrt{5} \neq 6 \sqrt{3}\)[/tex], this option is not valid.
Given the analysis above, none of the provided sets of values could be the side lengths of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle. Therefore, the answer is:
[tex]\[ \boxed{\text{None}} \][/tex]
