To determine which of the given ratios could represent the ratio of the length of the longer leg of a 30-60-90 triangle to the length of its hypotenuse, let's use the properties of a 30-60-90 triangle.
### Properties of a 30-60-90 Triangle
In a 30-60-90 triangle:
- The ratio of the shorter leg to the longer leg to the hypotenuse is [tex]\(1 : \sqrt{3} : 2\)[/tex].
- The ratio of the longer leg to the hypotenuse is therefore [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
We need to find which of the given ratios matches [tex]\(\frac{\sqrt{3}}{2}\)[/tex]:
1. Option A: [tex]\(1 : \sqrt{3}\)[/tex]
- Ratio: [tex]\(\frac{1}{\sqrt{3}}\)[/tex]
- Simplified: [tex]\(\frac{1}{\sqrt{3}} \neq \frac{\sqrt{3}}{2}\)[/tex]
- Not a valid ratio.
2. Option B: [tex]\(3 : 2 \sqrt{3}\)[/tex]
- Ratio: [tex]\(\frac{3}{2 \sqrt{3}}\)[/tex]
- Simplified: [tex]\(\frac{3}{2 \sqrt{3}} = \frac{3}{2} \cdot \frac{1}{\sqrt{3}} = \frac{3}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2}\)[/tex]
- This is a valid ratio.
3. Option C: [tex]\(\sqrt{3} : 2\)[/tex]
- Ratio: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- This exactly matches [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- This is a valid ratio.
4. Option D: [tex]\(\sqrt{3} : \sqrt{3}\)[/tex]
- Ratio: [tex]\(\frac{\sqrt{3}}{\sqrt{3}}\)[/tex]
- Simplified: [tex]\(\frac{\sqrt{3}}{\sqrt{3}} = 1 \neq \frac{\sqrt{3}}{2}\)[/tex]
- Not a valid ratio.
5. Option E: [tex]\(\sqrt{2} \cdot \sqrt{3} : 1\)[/tex]
- Ratio: [tex]\(\frac{\sqrt{2} \cdot \sqrt{3}}{1} = \sqrt{6}\)[/tex]
- [tex]\(\sqrt{6} \neq \frac{\sqrt{3}}{2}\)[/tex]
- Not a valid ratio.
6. Option F: [tex]\(3 \sqrt{3} : 6\)[/tex]
- Ratio: [tex]\(\frac{3 \sqrt{3}}{6}\)[/tex]
- Simplified: [tex]\(\frac{3 \sqrt{3}}{6} = \frac{\sqrt{3}}{2}\)[/tex]
- This is a valid ratio.
### Conclusion
The options that could represent the ratio of the length of the longer leg to the length of the hypotenuse in a 30-60-90 triangle are:
- B. [tex]\(3: 2 \sqrt{3}\)[/tex]
- C. [tex]\(\sqrt{3}: 2\)[/tex]
- F. [tex]\(3 \sqrt{3}: 6\)[/tex]
