To determine which of the given ratios could be the lengths of the two legs of a 30-60-90 triangle, we need to recall the properties of such a triangle. In a 30-60-90 triangle, the sides are in the specific ratio [tex]\(1:\sqrt{3}:2\)[/tex]. This means:
- The shortest leg (opposite the 30° angle) has length [tex]\(x\)[/tex].
- The longer leg (opposite the 60° angle) has length [tex]\(x\sqrt{3}\)[/tex].
- The hypotenuse (opposite the 90° angle) has length [tex]\(2x\)[/tex].
Let’s analyze each option:
A. [tex]\(\sqrt{2}: \sqrt{2}\)[/tex]
- This simplifies to [tex]\(1:1\)[/tex], which is not the ratio between the legs in a 30-60-90 triangle.
B. [tex]\(1: \sqrt{3}\)[/tex]
- This ratio matches the characteristic ratio between the shorter leg and the longer leg of a 30-60-90 triangle, as [tex]\(x:x\sqrt{3}\)[/tex].
c. [tex]\(\sqrt{3}: \sqrt{3}\)[/tex]
- This simplifies to [tex]\(1:1\)[/tex], which is not the ratio between the legs in a 30-60-90 triangle.
D. [tex]\(1: \sqrt{2}\)[/tex]
- This does not resemble any ratio specific to a 30-60-90 triangle.
E. [tex]\(\sqrt{2}: \sqrt{3}\)[/tex]
- This ratio does not match the characteristic ratio of [tex]\(1:\sqrt{3}\)[/tex] for the two legs of a 30-60-90 triangle.
F. [tex]\(2 \sqrt{3}: 6\)[/tex]
- Simplify [tex]\(2 \sqrt{3}: 6\)[/tex]:
[tex]\[
2 \sqrt{3}: 6 = \frac{2 \sqrt{3}}{6} = \frac{\sqrt{3}}{3} = \sqrt{3}:3
\][/tex]
This simplifies to a ratio that is not a characteristic length ratio [tex]\((1:\sqrt{3})\)[/tex] of the two legs of a 30-60-90 triangle.
Only option B, [tex]\(1:\sqrt{3}\)[/tex], matches the characteristic ratio between the lengths of the two legs of a 30-60-90 triangle.
Therefore, the correct answer is:
[tex]\[
\boxed{\text{B}}
\][/tex]
