Answer :
In a 30-60-90 triangle, the sides are in a specific ratio:
- The side opposite the 30° angle (the shortest side) is adjacent to the 60° angle.
- The side opposite the 60° angle is [tex]\(\sqrt{3}\)[/tex] times the length of the shortest side.
- The side opposite the 90° angle (the hypotenuse) is 2 times the length of the shortest side.
Therefore, the ratio of the lengths of the two legs (the sides opposite the 30° and 60° angles) is [tex]$1: \sqrt{3}$[/tex] or [tex]\(\sqrt{3}: 1\)[/tex].
Let's analyze each option:
### Option A: [tex]\(3 : 3 \sqrt{3}\)[/tex]
- Simplify the ratio: [tex]\(\frac{3}{3 \sqrt{3}} = \frac{1}{\sqrt{3}}\)[/tex]
- This matches the given ratio [tex]\(1 : \sqrt{3}\)[/tex].
- Therefore, Option A is correct.
### Option B: [tex]\(1 : \sqrt{2}\)[/tex]
- Simplify the ratio: [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
- This does not match [tex]\(1 : \sqrt{3}\)[/tex].
- Therefore, Option B is incorrect.
### Option C: [tex]\(\sqrt{2} : \sqrt{2}\)[/tex]
- Simplify the ratio: [tex]\(\frac{\sqrt{2}}{\sqrt{2}} = 1\)[/tex]
- This does not match [tex]\(1 : \sqrt{3}\)[/tex].
- Therefore, Option C is incorrect.
### Option D: [tex]\(1 : \sqrt{3}\)[/tex]
- This is already in the form [tex]\(1 : \sqrt{3}\)[/tex].
- Therefore, Option D is correct.
### Option E: [tex]\(\sqrt{2} : \sqrt{3}\)[/tex]
- Simplify the ratio: [tex]\(\frac{\sqrt{2}}{\sqrt{3}}\)[/tex]
- This does not match [tex]\(1 : \sqrt{3}\)[/tex].
- Therefore, Option E is incorrect.
### Option F: [tex]\(\sqrt{3} : 3\)[/tex]
- Simplify the ratio: [tex]\(\frac{\sqrt{3}}{3} = \frac{1}{\sqrt{3}}\)[/tex]
- This matches the given ratio [tex]\(1 : \sqrt{3}\)[/tex].
- Therefore, Option F is correct.
### Conclusion:
The ratios between the lengths of the two legs of a 30-60-90 triangle that match are:
- Option A: [tex]\(3 : 3 \sqrt{3}\)[/tex]
- Option D: [tex]\(1 : \sqrt{3}\)[/tex]
- Option F: [tex]\(\sqrt{3} : 3\)[/tex]
Thus, the correct answers are A, D, and F.
- The side opposite the 30° angle (the shortest side) is adjacent to the 60° angle.
- The side opposite the 60° angle is [tex]\(\sqrt{3}\)[/tex] times the length of the shortest side.
- The side opposite the 90° angle (the hypotenuse) is 2 times the length of the shortest side.
Therefore, the ratio of the lengths of the two legs (the sides opposite the 30° and 60° angles) is [tex]$1: \sqrt{3}$[/tex] or [tex]\(\sqrt{3}: 1\)[/tex].
Let's analyze each option:
### Option A: [tex]\(3 : 3 \sqrt{3}\)[/tex]
- Simplify the ratio: [tex]\(\frac{3}{3 \sqrt{3}} = \frac{1}{\sqrt{3}}\)[/tex]
- This matches the given ratio [tex]\(1 : \sqrt{3}\)[/tex].
- Therefore, Option A is correct.
### Option B: [tex]\(1 : \sqrt{2}\)[/tex]
- Simplify the ratio: [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
- This does not match [tex]\(1 : \sqrt{3}\)[/tex].
- Therefore, Option B is incorrect.
### Option C: [tex]\(\sqrt{2} : \sqrt{2}\)[/tex]
- Simplify the ratio: [tex]\(\frac{\sqrt{2}}{\sqrt{2}} = 1\)[/tex]
- This does not match [tex]\(1 : \sqrt{3}\)[/tex].
- Therefore, Option C is incorrect.
### Option D: [tex]\(1 : \sqrt{3}\)[/tex]
- This is already in the form [tex]\(1 : \sqrt{3}\)[/tex].
- Therefore, Option D is correct.
### Option E: [tex]\(\sqrt{2} : \sqrt{3}\)[/tex]
- Simplify the ratio: [tex]\(\frac{\sqrt{2}}{\sqrt{3}}\)[/tex]
- This does not match [tex]\(1 : \sqrt{3}\)[/tex].
- Therefore, Option E is incorrect.
### Option F: [tex]\(\sqrt{3} : 3\)[/tex]
- Simplify the ratio: [tex]\(\frac{\sqrt{3}}{3} = \frac{1}{\sqrt{3}}\)[/tex]
- This matches the given ratio [tex]\(1 : \sqrt{3}\)[/tex].
- Therefore, Option F is correct.
### Conclusion:
The ratios between the lengths of the two legs of a 30-60-90 triangle that match are:
- Option A: [tex]\(3 : 3 \sqrt{3}\)[/tex]
- Option D: [tex]\(1 : \sqrt{3}\)[/tex]
- Option F: [tex]\(\sqrt{3} : 3\)[/tex]
Thus, the correct answers are A, D, and F.
