A 30-60-90 triangle has specific side ratios based on its angles. The sides of a 30-60-90 triangle have the ratio:
1. The shortest leg (opposite the 30° angle) is [tex]\( \frac{1}{2} \)[/tex] the hypotenuse.
2. The longer leg (opposite the 60° angle) is [tex]\( \frac{\sqrt{3}}{2} \)[/tex] times the hypotenuse.
3. The hypotenuse is the longest side.
Given the properties of a 30-60-90 triangle, let's determine the ratio of the longer leg to the hypotenuse.
The ratio of the longer leg to the hypotenuse in a 30-60-90 triangle is:
[tex]\[ \frac{\frac{\sqrt{3}}{2}}{1} = \frac{\sqrt{3}}{2}. \][/tex]
We need to examine each given option to see if it simplifies to [tex]\( \frac{\sqrt{3}}{2} \)[/tex].
### Option A: [tex]\( \frac{3\sqrt{3}}{6} \)[/tex]
Simplify:
[tex]\[ \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2}. \][/tex]
This matches [tex]\( \frac{\sqrt{3}}{2} \)[/tex].
### Option B: [tex]\( \sqrt{3}:\sqrt{3} \)[/tex]
Simplify:
[tex]\[ \frac{\sqrt{3}}{\sqrt{3}} = 1. \][/tex]
This does not match [tex]\( \frac{\sqrt{3}}{2} \)[/tex].
### Option C: [tex]\( \frac{\sqrt{3}}{2} \)[/tex]
This directly matches [tex]\( \frac{\sqrt{3}}{2} \)[/tex].
### Option D: [tex]\( \frac{3}{2\sqrt{3}} \)[/tex]
Simplify:
[tex]\[ \frac{3}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2}. \][/tex]
This matches [tex]\( \frac{\sqrt{3}}{2} \)[/tex].
### Option E: [tex]\( \frac{1}{\sqrt{3}} \)[/tex]
Simplify:
[tex]\[ \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}. \][/tex]
This does not match [tex]\( \frac{\sqrt{3}}{2} \)[/tex].
### Option F: [tex]\( \frac{\sqrt{2}}{\sqrt{3}} \)[/tex]
Simplify:
[tex]\[ \frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{6}}{3}. \][/tex]
This does not match [tex]\( \frac{\sqrt{3}}{2} \)[/tex].
In conclusion, the options that could be the ratio of the length of the longer leg of a 30-60-90 triangle to the length of its hypotenuse are:
- A: [tex]\( \frac{3\sqrt{3}}{6} \)[/tex]
- C: [tex]\( \frac{\sqrt{3}}{2} \)[/tex]
- D: [tex]\( \frac{3}{2\sqrt{3}} \)[/tex]
