To determine which of the given options could be the ratio of the length of the longer leg of a 30-60-90 triangle to the length of its hypotenuse, we can start by recalling the properties of a 30-60-90 triangle.
In a 30-60-90 triangle:
- The ratio of the shorter leg to the longer leg is [tex]\(1:\sqrt{3}\)[/tex].
- The ratio of the shorter leg to the hypotenuse is [tex]\(1:2\)[/tex].
- Hence, the ratio of the longer leg to the hypotenuse is [tex]\(\sqrt{3}:2\)[/tex].
Now let's check each given option to see if it matches the ratio [tex]\(\sqrt{3}:2\)[/tex]:
### A. [tex]\(\sqrt{3}:2\)[/tex]
This matches the exact ratio required. Therefore, A is a correct option.
### B. [tex]\(3 \sqrt{3}:6\)[/tex]
To simplify [tex]\(3 \sqrt{3}:6\)[/tex]:
[tex]\[
\frac{3 \sqrt{3}}{6} = \frac{ \sqrt{3}}{2}
\][/tex]
This matches the ratio required. Therefore, B is a correct option.
### C. [tex]\(1:\sqrt{3}\)[/tex]
To see if [tex]\(1:\sqrt{3}\)[/tex] matches the ratio [tex]\(\sqrt{3}:2\)[/tex], simplify [tex]\(1:\sqrt{3}\)[/tex]:
[tex]\[
\frac{1}{\sqrt{3}} \neq \frac{\sqrt{3}}{2}
\][/tex]
Therefore, C is not a correct option.
### D. [tex]\(\sqrt{2}:\sqrt{3}\)[/tex]
To see if [tex]\(\sqrt{2}:\sqrt{3}\)[/tex] matches the ratio [tex]\(\sqrt{3}:2\)[/tex], simplify [tex]\(\sqrt{2}:\sqrt{3}\)[/tex]:
[tex]\[
\frac{\sqrt{2}}{\sqrt{3}} = \frac{\sqrt{2 \cdot 3}}{3} = \frac{\sqrt{6}}{3} \neq \frac{\sqrt{3}}{2}
\][/tex]
Therefore, D is not a correct option.
### E. [tex]\(\sqrt{3}:\sqrt{3}\)[/tex]
To see if [tex]\(\sqrt{3}:\sqrt{3}\)[/tex] matches the ratio [tex]\(\sqrt{3}:2\)[/tex], simplify [tex]\(\sqrt{3}:\sqrt{3}\)[/tex]:
[tex]\[
\frac{\sqrt{3}}{\sqrt{3}} = 1 \neq \frac{\sqrt{3}}{2}
\][/tex]
Therefore, E is not a correct option.
### F. [tex]\(3:2 \sqrt{3}\)[/tex]
To see if [tex]\(3:2\sqrt{3}\)[/tex] matches the ratio [tex]\(\sqrt{3}:2\)[/tex], simplify [tex]\(3:2\sqrt{3}\)[/tex]:
[tex]\[
\frac{3}{2 \sqrt{3}} = \frac{3 \sqrt{3}}{6} = \frac{\sqrt{3}}{2}
\][/tex]
This matches the ratio required. Therefore, F is a correct option.
### Conclusion
The correct options that match the ratio of the length of the longer leg of a 30-60-90 triangle to the length of its hypotenuse are:
A. [tex]\(\sqrt{3}:2\)[/tex]
B. [tex]\(3 \sqrt{3}:6\)[/tex]
F. [tex]\(3:2 \sqrt{3}\)[/tex]
