To solve this problem, we need to understand the inherent ratios of the sides of a 30-60-90 triangle. In a 30-60-90 triangle, the sides are proportional to specific values based on the angles:
- The side opposite the 30° angle is the shortest and is denoted by [tex]\(x\)[/tex].
- The side opposite the 60° angle is the longer leg and has a length of [tex]\(x \sqrt{3}\)[/tex].
- The hypotenuse (opposite the 90° angle) is the longest side and has a length of [tex]\(2x\)[/tex].
We seek the ratio of the length of the longer leg to the hypotenuse, which would be:
[tex]\[
\frac{length \, of \, longer \, leg}{length \, of \, hypotenuse} = \frac{x \sqrt{3}}{2x} = \frac{\sqrt{3}}{2}
\][/tex]
Let's check each option to see which ones match this ratio ([tex]\(\sqrt{3} : 2\)[/tex]):
A. [tex]\(3: 2 \sqrt{3}\)[/tex]
[tex]\[
\frac{3}{2 \sqrt{3}}
\][/tex]
Simplifying this, we get:
[tex]\[
\frac{3}{2 \sqrt{3}} = \frac{3}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{3 \sqrt{3}}{6} = \frac{\sqrt{3}}{2}
\][/tex]
This matches the required ratio.
B. 1: [tex]\(\sqrt{3}\)[/tex]
[tex]\[
\frac{1}{\sqrt{3}}
\][/tex]
This ratio does not simplify to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
C. [tex]\(\sqrt{2}: \sqrt{3}\)[/tex]
[tex]\[
\frac{\sqrt{2}}{\sqrt{3}}
\][/tex]
This ratio does not simplify to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
D. [tex]\(\sqrt{3}: 2\)[/tex]
[tex]\[
\frac{\sqrt{3}}{2}
\][/tex]
This directly matches the required ratio.
E. [tex]\(\sqrt{3}: \sqrt{3}\)[/tex]
[tex]\[
\frac{\sqrt{3}}{\sqrt{3}} = 1
\][/tex]
This does not match the required ratio.
F. [tex]\(3 \sqrt{3}: 6\)[/tex]
[tex]\[
\frac{3 \sqrt{3}}{6} = \frac{\sqrt{3}}{2}
\][/tex]
This matches the required ratio.
Thus, the correct answer choices are:
A. [tex]\(3: 2 \sqrt{3}\)[/tex]
D. [tex]\(\sqrt{3}: 2\)[/tex]
F. [tex]\(3 \sqrt{3}: 6\)[/tex]
