Answer :
To determine which ratios could describe the relationship between the lengths of the two legs of a 30-60-90 triangle, we need to recall the properties of this special kind of right triangle. In a 30-60-90 triangle, the side lengths are in the ratio [tex]\(1:\sqrt{3}:2\)[/tex], where:
- The side opposite the 30° angle has a length of [tex]\(x\)[/tex]
- The side opposite the 60° angle has a length of [tex]\(x\sqrt{3}\)[/tex]
- The hypotenuse has a length of [tex]\(2x\)[/tex]
Given the side opposite the 30° angle is [tex]\(x\)[/tex] and the side opposite the 60° angle is [tex]\(x\sqrt{3}\)[/tex], the ratio of the two legs (the sides opposite the 30° and 60° angles) is:
[tex]\[ \text{Ratio} = \frac{x}{x\sqrt{3}} = \frac{1}{\sqrt{3}} \][/tex]
Now, let's check each given ratio to see if it is equivalent to [tex]\( \frac{1}{\sqrt{3}} \)[/tex]:
A. [tex]\( \sqrt{2}: \sqrt{3} \)[/tex]
[tex]\[ \text{Ratio} = \frac{\sqrt{2}}{\sqrt{3}} \neq \frac{1}{\sqrt{3}} \][/tex]
This is not a valid ratio.
B. [tex]\( 2 \sqrt{3}: 6 \)[/tex]
[tex]\[ \text{Ratio} = \frac{2\sqrt{3}}{6} = \frac{\sqrt{3}}{3} = \frac{\sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{\sqrt{9}} = \frac{\sqrt{3}}{3} \][/tex]
Simplifying we see:
[tex]\[ \frac{\sqrt{3}}{3} = \frac{1}{\sqrt{3}} \][/tex]
This is a valid ratio.
C. [tex]\( 1: \sqrt{3} \)[/tex]
[tex]\[ \text{Ratio} = \frac{1}{\sqrt{3}} \][/tex]
This is already in the form [tex]\(\frac{1}{\sqrt{3}}\)[/tex], so it is valid.
D. [tex]\( 1: \sqrt{2} \)[/tex]
[tex]\[ \text{Ratio} = \frac{1}{\sqrt{2}} \neq \frac{1}{\sqrt{3}} \][/tex]
This is not a valid ratio.
E. [tex]\( \sqrt{2}: \sqrt{2} \)[/tex]
[tex]\[ \text{Ratio} = \frac{\sqrt{2}}{\sqrt{2}} = 1 \neq \frac{1}{\sqrt{3}} \][/tex]
This is not a valid ratio.
F. [tex]\( \sqrt{3}: \sqrt{3} \)[/tex]
[tex]\[ \text{Ratio} = \frac{\sqrt{3}}{\sqrt{3}} = 1 \neq \frac{1}{\sqrt{3}} \][/tex]
This is not a valid ratio.
Thus, the ratios that could describe the lengths of the two legs of a 30-60-90 triangle are:
[tex]\[ \boxed{B, C, E} \][/tex]
- The side opposite the 30° angle has a length of [tex]\(x\)[/tex]
- The side opposite the 60° angle has a length of [tex]\(x\sqrt{3}\)[/tex]
- The hypotenuse has a length of [tex]\(2x\)[/tex]
Given the side opposite the 30° angle is [tex]\(x\)[/tex] and the side opposite the 60° angle is [tex]\(x\sqrt{3}\)[/tex], the ratio of the two legs (the sides opposite the 30° and 60° angles) is:
[tex]\[ \text{Ratio} = \frac{x}{x\sqrt{3}} = \frac{1}{\sqrt{3}} \][/tex]
Now, let's check each given ratio to see if it is equivalent to [tex]\( \frac{1}{\sqrt{3}} \)[/tex]:
A. [tex]\( \sqrt{2}: \sqrt{3} \)[/tex]
[tex]\[ \text{Ratio} = \frac{\sqrt{2}}{\sqrt{3}} \neq \frac{1}{\sqrt{3}} \][/tex]
This is not a valid ratio.
B. [tex]\( 2 \sqrt{3}: 6 \)[/tex]
[tex]\[ \text{Ratio} = \frac{2\sqrt{3}}{6} = \frac{\sqrt{3}}{3} = \frac{\sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{\sqrt{9}} = \frac{\sqrt{3}}{3} \][/tex]
Simplifying we see:
[tex]\[ \frac{\sqrt{3}}{3} = \frac{1}{\sqrt{3}} \][/tex]
This is a valid ratio.
C. [tex]\( 1: \sqrt{3} \)[/tex]
[tex]\[ \text{Ratio} = \frac{1}{\sqrt{3}} \][/tex]
This is already in the form [tex]\(\frac{1}{\sqrt{3}}\)[/tex], so it is valid.
D. [tex]\( 1: \sqrt{2} \)[/tex]
[tex]\[ \text{Ratio} = \frac{1}{\sqrt{2}} \neq \frac{1}{\sqrt{3}} \][/tex]
This is not a valid ratio.
E. [tex]\( \sqrt{2}: \sqrt{2} \)[/tex]
[tex]\[ \text{Ratio} = \frac{\sqrt{2}}{\sqrt{2}} = 1 \neq \frac{1}{\sqrt{3}} \][/tex]
This is not a valid ratio.
F. [tex]\( \sqrt{3}: \sqrt{3} \)[/tex]
[tex]\[ \text{Ratio} = \frac{\sqrt{3}}{\sqrt{3}} = 1 \neq \frac{1}{\sqrt{3}} \][/tex]
This is not a valid ratio.
Thus, the ratios that could describe the lengths of the two legs of a 30-60-90 triangle are:
[tex]\[ \boxed{B, C, E} \][/tex]
