Let's delve into the properties of a 30-60-90 triangle to determine which of the given ratios could be the lengths of the two legs.
1. Understanding 30-60-90 Triangles:
A 30-60-90 triangle is a special right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees. The sides of this triangle have a fixed ratio:
- The side opposite the 30-degree angle (the shorter leg) can be denoted as [tex]\(x\)[/tex].
- The side opposite the 60-degree angle (the longer leg) is [tex]\(\sqrt{3}\)[/tex] times the shorter leg, so it is [tex]\(x\sqrt{3}\)[/tex].
- The side opposite the 90-degree angle (the hypotenuse) is twice the shorter leg, so it is [tex]\(2x\)[/tex].
2. Ratio of the Two Legs:
The two legs of the triangle are the sides opposite the 30-degree and 60-degree angles, which are [tex]\(x\)[/tex] and [tex]\(x\sqrt{3}\)[/tex] respectively. The ratio of these legs (shorter leg to longer leg) is:
[tex]\[
x : x\sqrt{3} = 1 : \sqrt{3}
\][/tex]
Let's check each given option against this correct ratio:
- Option A: [tex]\(1 : \sqrt{3}\)[/tex]
- This correctly matches the ratio of the two legs of a 30-60-90 triangle.
- Option B: [tex]\(\sqrt{3} : 3\)[/tex]
- This does not match the required ratio since it simplifies to [tex]\(\frac{\sqrt{3}}{3} : 1\)[/tex], which is not equivalent to [tex]\(1 : \sqrt{3}\)[/tex].
- Option C: [tex]\(\sqrt{2} : \sqrt{3}\)[/tex]
- This does not match the required ratio of [tex]\(1 : \sqrt{3}\)[/tex].
- Option D: [tex]\(\sqrt{3} : \sqrt{3}\)[/tex]
- This simplifies to [tex]\(1 : 1\)[/tex], which does not match [tex]\(1 : \sqrt{3}\)[/tex].
- Option E: [tex]\(1 : \sqrt{2}\)[/tex]
- This does not match the required ratio of [tex]\(1 : \sqrt{3}\)[/tex].
- Option F: [tex]\(\sqrt{2} : \sqrt{2}\)[/tex]
- This simplifies to [tex]\(1 : 1\)[/tex], which does not match [tex]\(1 : \sqrt{3}\)[/tex].
So, only option A, [tex]\(1 : \sqrt{3}\)[/tex], is the correct ratio between the lengths of the two legs of a 30-60-90 triangle.
Thus, the answer is:
[tex]\[ \boxed{A} \][/tex]
