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 first recall the properties of a 30-60-90 triangle. In such a triangle, the sides are in a specific ratio relative to each other:
- The hypotenuse is twice the length of the shorter leg.
- The length of the longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg.
Let’s test each option to see if it matches the required ratio.
A. [tex]\(2 \sqrt{3}: 6\)[/tex]
[tex]\[
\frac{2\sqrt{3}}{6} = \frac{2 \cdot \sqrt{3}}{2 \cdot 3} = \frac{\sqrt{3}}{3} \quad \text{(This ratio does not simplify to } 1:\sqrt{3} \text{)}
\][/tex]
So, option A is incorrect.
B. [tex]\(\sqrt{2}: \sqrt{3}\)[/tex]
[tex]\[
\text{This does not match the } 1: \sqrt{3} \text{ ratio since we need the shorter leg and the longer leg to be in a specific ratio of } 1:\sqrt{3}.
\][/tex]
So, option B is incorrect.
C. [tex]\(1: \sqrt{3}\)[/tex]
[tex]\[
\text{This ratio matches exactly with the required ratio of the lengths of the shorter leg to the longer leg in a } 30-60-90 \text{ triangle}.
\][/tex]
So, option C is correct.
D. [tex]\(1: \sqrt{2}\)[/tex]
[tex]\[
\text{This does not match the required ratio of } 1: \sqrt{3}.
\][/tex]
So, option D is incorrect.
E. [tex]\(\sqrt{2}: \sqrt{2}\)[/tex]
[tex]\[
\frac{\sqrt{2}}{\sqrt{2}} = 1:1 \quad \text{(This is clearly not the ratio between the side lengths of a } 30-60-90 \text{ triangle)}.
\][/tex]
So, option E is incorrect.
Thus, the only correct option is:
C. [tex]\(1: \sqrt{3}\)[/tex]
