To determine which of the given options could be the ratio of the length of the longer leg to the hypotenuse in a 30-60-90 triangle, we first need to understand the properties of such a triangle.
In a 30-60-90 triangle, the sides are in specific ratios:
- The hypotenuse is twice as long as the shorter leg.
- The longer leg is [tex]\( \sqrt{3} \)[/tex] times the length of the shorter leg.
Thus, the ratio of the longer leg to the hypotenuse in a 30-60-90 triangle is:
[tex]\[ \frac{\text{longer leg}}{\text{hypotenuse}} = \frac{\sqrt{3} \times \text{shorter leg}}{2 \times \text{shorter leg}} = \frac{\sqrt{3}}{2} \][/tex]
Now, let's analyze each option to see if it can be simplified to [tex]\(\frac{\sqrt{3}}{2}\)[/tex]:
A. [tex]\(\sqrt{3}: 2\)[/tex]
- This matches the ratio of the longer leg to the hypotenuse in a 30-60-90 triangle.
B. [tex]\(\sqrt{2}: \sqrt{3}\)[/tex]
- This is not the given ratio [tex]\(\frac{\sqrt{3}}{2}\)[/tex]. This ratio cannot be simplified to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
C. [tex]\(2 \sqrt{3}: 4\)[/tex]
- Simplify this ratio: [tex]\( \frac{2 \sqrt{3}}{4} = \frac{\sqrt{3}}{2} \)[/tex]
- This matches the required ratio.
D. [tex]\(\sqrt{3}: \sqrt{3}\)[/tex]
- This simplifies to [tex]\(1:1\)[/tex], not [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
E. [tex]\(1: \sqrt{3}\)[/tex]
- This does not match the ratio [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
F. [tex]\(2: 2 \sqrt{2}\)[/tex]
- Simplify this ratio: [tex]\( \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \)[/tex]
- This does not match the ratio [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Therefore, the correct ratios are:
- A. [tex]\(\sqrt{3}: 2\)[/tex]
- C. [tex]\(2 \sqrt{3}: 4\)[/tex]
Thus, the corresponding indices of these options are:
[tex]\[ \boxed{1 \text{ and } 3} \][/tex]
