To determine which of the provided ratios could represent the ratio of the length of the longer leg to the length of the hypotenuse in a 30-60-90 triangle, we need to understand the properties of a 30-60-90 triangle.
In a 30-60-90 triangle:
- The sides are in the ratio \(1 : \sqrt{3} : 2\),
where:
- \(1\) is the length of the shorter leg,
- \(\sqrt{3}\) is the length of the longer leg,
- \(2\) is the length of the hypotenuse.
Thus, the ratio of the longer leg to the hypotenuse is \(\frac{\sqrt{3}}{2}\).
Now, let's check each of the given options to determine if they are equivalent to \(\frac{\sqrt{3}}{2}\).
A. \(1 : \sqrt{3}\)
[tex]\[ \frac{1}{\sqrt{3}} \neq \frac{\sqrt{3}}{2} \][/tex]
Hence, this is incorrect.
B. \(\sqrt{3} : 2\)
[tex]\[ \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \][/tex]
This is already in the desired form.
C. \(2 : 2\sqrt{2}\)
[tex]\[ \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \][/tex]
This is not equivalent to \(\frac{\sqrt{3}}{2}\).
D. \(\sqrt{3} : \sqrt{3}\)
[tex]\[ \frac{\sqrt{3}}{\sqrt{3}} = 1 \][/tex]
This is not equivalent to \(\frac{\sqrt{3}}{2}\).
E. \(2\sqrt{3} : 4\)
[tex]\[ \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \][/tex]
This is equivalent to \(\frac{\sqrt{3}}{2}\).
F. \(\sqrt{2} : \sqrt{3}\)
[tex]\[ \frac{\sqrt{2}}{\sqrt{3}} \neq \frac{\sqrt{3}}{2} \][/tex]
This is not equivalent.
Thus, the ratios that represent the ratio of the length of the longer leg to the length of the hypotenuse in a 30-60-90 triangle are:
[tex]\[ \boxed{B, E} \][/tex]
