To solve the problem, we need to understand the properties of a 30-60-90 triangle. A 30-60-90 triangle is a special type of right triangle where the angles are exactly 30 degrees, 60 degrees, and 90 degrees. The sides of such a triangle have fixed ratios relative to each other:
1. The side opposite the 30-degree angle is the shortest and is called the "short leg."
2. The side opposite the 60-degree angle is the "long leg."
3. The side opposite the 90-degree angle is the hypotenuse.
The relationships between these sides are:
- The short leg is exactly half the hypotenuse.
- The long leg is the short leg multiplied by [tex]\(\sqrt{3}\)[/tex].
If we denote the short leg by [tex]\(a\)[/tex]:
- The hypotenuse will be [tex]\(2a\)[/tex].
- The long leg will be [tex]\(a\sqrt{3}\)[/tex].
Thus, the ratio between the lengths of the short leg and the long leg in a 30-60-90 triangle is:
[tex]\[ 1 : \sqrt{3} \][/tex]
Now let's analyze the given options to find which ones match this ratio:
A. [tex]\(\sqrt{3}: 3\)[/tex]
- The ratio [tex]\(\frac{\sqrt{3}}{3}\)[/tex] simplifies to [tex]\(\frac{1}{\sqrt{3}}\)[/tex]. This does not match [tex]\(1 : \sqrt{3}\)[/tex].
B. [tex]\(1: \sqrt{2}\)[/tex]
- The ratio [tex]\(1: \sqrt{2}\)[/tex] does not match [tex]\(1: \sqrt{3}\)[/tex].
C. [tex]\(1: \sqrt{3}\)[/tex]
- This exactly matches the ratio [tex]\(1 : \sqrt{3}\)[/tex].
D. [tex]\(\sqrt{3}: \sqrt{3}\)[/tex]
- The ratio [tex]\(\sqrt{3} : \sqrt{3}\)[/tex] simplifies to [tex]\(1:1\)[/tex], which does not match [tex]\(1 : \sqrt{3}\)[/tex].
E. [tex]\(\sqrt{2} : \sqrt{3}\)[/tex]
- The ratio [tex]\(\frac{\sqrt{2}}{\sqrt{3}}\)[/tex] does not match [tex]\(1 : \sqrt{3}\)[/tex].
F. [tex]\(\sqrt{2} : \sqrt{2}\)[/tex]
- The ratio [tex]\(\sqrt{2} : \sqrt{2}\)[/tex] simplifies to [tex]\(1:1\)[/tex], which does not match [tex]\(1 : \sqrt{3}\)[/tex].
Therefore, the only correct ratio that matches the one in a 30-60-90 triangle from the provided options is:
[tex]\[ \boxed{C} \][/tex]
