To determine which of the given ratios could be the ratio between the lengths of the two legs of a 30-60-90 triangle, let's recall the properties of a 30-60-90 triangle. For such a triangle, the sides are in the ratio [tex]\(1 : \sqrt{3} : 2\)[/tex], where the shortest side is opposite the 30 degrees, the side opposite the 60 degrees is [tex]\(\sqrt{3}\)[/tex] times the shortest side, and the hypotenuse is 2 times the shortest side.
Now, let's analyze each given option to see if the ratio matches [tex]\(1 : \sqrt{3}\)[/tex].
Option A: [tex]\(\sqrt{2} : \sqrt{3}\)[/tex]
The ratio here is [tex]\(\sqrt{2} / \sqrt{3}\)[/tex].
If we simplify the ratio:
[tex]\[
\frac{\sqrt{2}}{\sqrt{3}} = \sqrt{\frac{2}{3}}
\][/tex]
This does not match the [tex]\(1 : \sqrt{3}\)[/tex] ratio expected in a 30-60-90 triangle.
Option B: [tex]\(\sqrt{3} : \sqrt{3}\)[/tex]
The ratio here is [tex]\(\sqrt{3} / \sqrt{3}\)[/tex], which simplifies to:
[tex]\[
\frac{\sqrt{3}}{\sqrt{3}} = 1
\][/tex]
This matches the [tex]\(1 : \sqrt{3}\)[/tex] ratio.
Option C: [tex]\(1 : \sqrt{2}\)[/tex]
The ratio here is [tex]\(1 / \sqrt{2}\)[/tex].
[tex]\[
\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\][/tex]
This does not match the [tex]\(1 : \sqrt{3}\)[/tex] ratio.
Option D: [tex]\(\sqrt{2} : \sqrt{2}\)[/tex]
The ratio here is [tex]\(\sqrt{2} / \sqrt{2}\)[/tex], which simplifies to:
[tex]\[
\frac{\sqrt{2}}{\sqrt{2}} = 1
\][/tex]
This matches the [tex]\(1 : \sqrt{3}\)[/tex] ratio.
Option E: [tex]\(1 : \sqrt{3}\)[/tex]
The ratio here is [tex]\(1 / \sqrt{3}\)[/tex]:
[tex]\[
\frac{1}{\sqrt{3}}
\][/tex]
This exactly matches the [tex]\(1 : \sqrt{3}\)[/tex] ratio.
Option F: [tex]\(2\sqrt{5} : 6\)[/tex]
The ratio here is [tex]\(2\sqrt{5} / 6\)[/tex]:
[tex]\[
\frac{2\sqrt{5}}{6} = \frac{\sqrt{5}}{3}
\][/tex]
This does not match the [tex]\(1 : \sqrt{3}\)[/tex] ratio.
Based on the above analysis, the ratios that match the ratio between the lengths of the two legs of a 30-60-90 triangle are:
- Option B: [tex]\(\sqrt{3} : \sqrt{3}\)[/tex]
- Option D: [tex]\(\sqrt{2} : \sqrt{2}\)[/tex]
- Option E: [tex]\(1 : \sqrt{3}\)[/tex]
Thus, the correct options are B, D, and E.
