Answer :
To determine which of the given options could be the ratio between the lengths of the two legs of a \(30^\circ-60^\circ-90^\circ\) triangle, we first need to recall the properties of this special right triangle. In a \(30^\circ-60^\circ-90^\circ\) triangle, the side lengths are in a specific ratio:
- The shorter leg (opposite the \(30^\circ\) angle) is \(a\).
- The longer leg (opposite the \(60^\circ\) angle) is \(a\sqrt{3}\).
- The hypotenuse (opposite the \(90^\circ\) angle) is \(2a\).
Given this information, we see that the ratio between the shorter leg and the longer leg is \(1 : \sqrt{3}\). We will now evaluate each option to see if it corresponds to this ratio.
### Option A: \(2 \sqrt{3}: 6\)
To simplify:
[tex]\[ \frac{2\sqrt{3}}{6} = \frac{\sqrt{3}}{3} = \frac{1}{\sqrt{3}} \][/tex]
This simplifies to \(1 : \sqrt{3}\), which is the correct ratio.
### Option B: \(1: \sqrt{3}\)
This option is already in the simplified form \(1 : \sqrt{3}\), which matches our required ratio directly.
### Option C: \(1: \sqrt{2}\)
This option is \(1 : \sqrt{2}\), which does not correspond to our required ratio of \(1 : \sqrt{3}\). Therefore, this is not a valid answer.
### Option D: \(\sqrt{2}: \sqrt{3}\)
To simplify:
[tex]\[ \frac{\sqrt{2}}{\sqrt{3}} \][/tex]
This does not simplify to \(1 : \sqrt{3}\) and thus does not correspond to the required ratio.
### Option E: \(\sqrt{3}: \sqrt{3}\)
To simplify:
[tex]\[ \frac{\sqrt{3}}{\sqrt{3}} = 1 \][/tex]
This ratio is \(1 : 1\), not \(1 : \sqrt{3}\), so it is not correct.
### Option F: \(\sqrt{2}: \sqrt{2}\)
To simplify:
[tex]\[ \frac{\sqrt{2}}{\sqrt{2}} = 1 \][/tex]
This ratio is \(1 : 1\), not \(1 : \sqrt{3}\), so it is not correct.
Based on the evaluation of each option, the correct answers are:
A. [tex]\(2 \sqrt{3}: 6\)[/tex] and B. [tex]\(1: \sqrt{3}\)[/tex].
- The shorter leg (opposite the \(30^\circ\) angle) is \(a\).
- The longer leg (opposite the \(60^\circ\) angle) is \(a\sqrt{3}\).
- The hypotenuse (opposite the \(90^\circ\) angle) is \(2a\).
Given this information, we see that the ratio between the shorter leg and the longer leg is \(1 : \sqrt{3}\). We will now evaluate each option to see if it corresponds to this ratio.
### Option A: \(2 \sqrt{3}: 6\)
To simplify:
[tex]\[ \frac{2\sqrt{3}}{6} = \frac{\sqrt{3}}{3} = \frac{1}{\sqrt{3}} \][/tex]
This simplifies to \(1 : \sqrt{3}\), which is the correct ratio.
### Option B: \(1: \sqrt{3}\)
This option is already in the simplified form \(1 : \sqrt{3}\), which matches our required ratio directly.
### Option C: \(1: \sqrt{2}\)
This option is \(1 : \sqrt{2}\), which does not correspond to our required ratio of \(1 : \sqrt{3}\). Therefore, this is not a valid answer.
### Option D: \(\sqrt{2}: \sqrt{3}\)
To simplify:
[tex]\[ \frac{\sqrt{2}}{\sqrt{3}} \][/tex]
This does not simplify to \(1 : \sqrt{3}\) and thus does not correspond to the required ratio.
### Option E: \(\sqrt{3}: \sqrt{3}\)
To simplify:
[tex]\[ \frac{\sqrt{3}}{\sqrt{3}} = 1 \][/tex]
This ratio is \(1 : 1\), not \(1 : \sqrt{3}\), so it is not correct.
### Option F: \(\sqrt{2}: \sqrt{2}\)
To simplify:
[tex]\[ \frac{\sqrt{2}}{\sqrt{2}} = 1 \][/tex]
This ratio is \(1 : 1\), not \(1 : \sqrt{3}\), so it is not correct.
Based on the evaluation of each option, the correct answers are:
A. [tex]\(2 \sqrt{3}: 6\)[/tex] and B. [tex]\(1: \sqrt{3}\)[/tex].
