To determine the correct ratios representing the length of the longer leg to the hypotenuse in a 30-60-90 triangle, we need to remember the fundamental properties of this special triangle. In a 30-60-90 triangle:
1. The shorter leg (opposite the 30° angle) is half the hypotenuse.
2. The longer leg (opposite the 60° angle) is [tex]\(\sqrt{3}\)[/tex] times the shorter leg.
Let's denote:
- [tex]\(a\)[/tex] as the shorter leg,
- [tex]\(h\)[/tex] as the hypotenuse,
- [tex]\(b\)[/tex] as the longer leg.
From the properties:
[tex]\[ h = 2a \][/tex]
[tex]\[ b = a\sqrt{3} \][/tex]
Thus, the ratio of the longer leg to the hypotenuse is:
[tex]\[ \frac{b}{h} = \frac{a\sqrt{3}}{2a} = \frac{\sqrt{3}}{2} \][/tex]
Hence, the ratio of the longer leg to the hypotenuse in a 30-60-90 triangle is [tex]\(\sqrt{3}:2\)[/tex].
Now let's analyze each option:
### Option A: [tex]\(\sqrt{2}: \sqrt{3}\)[/tex]
This does not simplify to [tex]\(\sqrt{3}: 2\)[/tex]. Therefore, it is not correct.
### Option B: [tex]\(1: \sqrt{3}\)[/tex]
This also does not simplify to [tex]\(\sqrt{3}: 2\)[/tex]. Therefore, it is not correct.
### Option C: [tex]\(\sqrt{3}: 2\)[/tex]
This is exactly the ratio we found for the longer leg to the hypotenuse in a 30-60-90 triangle. Therefore, it is correct.
### Option D: [tex]\(3\sqrt{3}: 6\)[/tex]
Simplifying this ratio:
[tex]\[ \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2} \][/tex]
This simplifies to [tex]\(\sqrt{3}: 2\)[/tex], so it is correct.
### Option E: [tex]\(3: 2\sqrt{3}\)[/tex]
Simplifying this ratio:
[tex]\[ \frac{3}{2\sqrt{3}} = \frac{3}{2} \cdot \frac{1}{\sqrt{3}} = \frac{3}{2} \cdot \frac{\sqrt{3}}{3} = \frac{\sqrt{3}}{2} \][/tex]
This simplifies to [tex]\(\sqrt{3}: 2\)[/tex], so it is correct.
### Option F: [tex]\(\sqrt{3}: \sqrt{3}\)[/tex]
This simplifies to [tex]\(1:1\)[/tex], which is not [tex]\(\sqrt{3}: 2\)[/tex]. Therefore, it is not correct.
Therefore, the correct options that represent the ratio of the longer leg to the hypotenuse in a 30-60-90 triangle are:
- C. [tex]\(\sqrt{3}: 2\)[/tex]
- D. [tex]\(3\sqrt{3}: 6\)[/tex]
- E. [tex]\(3: 2\sqrt{3}\)[/tex]
