To determine the possible ratios of the length of the longer leg (opposite the [tex]$60^\circ$[/tex] angle) to the length of the hypotenuse in a [tex]$30^\circ$[/tex]-[tex]$60^\circ$[/tex]-[tex]$90^\circ$[/tex] triangle, we need to remember the properties of such a triangle. In a [tex]$30^\circ$[/tex]-[tex]$60^\circ$[/tex]-[tex]$90^\circ$[/tex] triangle, the sides are in the ratio [tex]\(1 : \sqrt{3} : 2\)[/tex], where:
- The shortest side, opposite the [tex]$30^\circ$[/tex] angle, has length [tex]\(1\)[/tex].
- The longer leg, opposite the [tex]$60^\circ$[/tex] angle, has length [tex]\(\sqrt{3}\)[/tex].
- The hypotenuse has length [tex]\(2\)[/tex].
Now, we need to calculate the ratio of the length of the longer leg to the length of the hypotenuse:
[tex]\[
\text{Longer leg (opposite $60^\circ$)} : \text{Hypotenuse} = \frac{\sqrt{3}}{2}
\][/tex]
Let’s evaluate each option to see if it matches this ratio:
A. [tex]\( \frac{3}{2 \sqrt{3}} \)[/tex]
Simplify:
[tex]\[
\frac{3}{2 \sqrt{3}} = \frac{3}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{3 \sqrt{3}}{2 \cdot 3} = \frac{\sqrt{3}}{2}
\][/tex]
This matches [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
B. [tex]\( \frac{2}{2 \sqrt{2}} \)[/tex]
Simplify:
[tex]\[
\frac{2}{2 \sqrt{2}} = \frac{2}{2 \sqrt{2}} = \frac{2}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{2 \sqrt{2}}{4} = \frac{\sqrt{2}}{2}
\][/tex]
This does not match [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
C. [tex]\( \frac{\sqrt{3}}{2} \)[/tex]
This directly matches [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
D. [tex]\( \frac{1}{\sqrt{2}} \)[/tex]
Simplify:
[tex]\[
\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\][/tex]
This does not match [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
E. [tex]\( \frac{2}{3 \sqrt{3}} \)[/tex]
Simplify:
[tex]\[
\frac{2}{3 \sqrt{3}} = \frac{2}{3 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2 \sqrt{3}}{9} = \frac{2 \sqrt{3}}{9}
\][/tex]
This does not match [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
F. [tex]\( \frac{\sqrt{2}}{\sqrt{3}} \)[/tex]
Simplify:
[tex]\[
\frac{\sqrt{2}}{\sqrt{3}} = \frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{6}}{3}
\][/tex]
This does not match [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
So, the only ratios that match the correct ratio [tex]\(\frac{\sqrt{3}}{2}\)[/tex] are:
A. [tex]\( \frac{3}{2 \sqrt{3}} \)[/tex] and C. [tex]\( \frac{\sqrt{3}}{2} \)[/tex].
The correct options are therefore:
[tex]\[
\boxed{A \text{ and } C}
\][/tex]
