Answer :
When dealing with a 30-60-90 triangle, it's essential to know the specific properties and side ratios of this type of special right triangle. A 30-60-90 triangle has the sides in the ratio:
- Opposite the 30-degree angle: 1
- Opposite the 60-degree angle: [tex]\(\sqrt{3}\)[/tex]
- Opposite the 90-degree angle (the hypotenuse): 2
To determine which of the given options could be the ratio between the lengths of the two legs (the sides opposite the 30-degree and 60-degree angles), we use the known ratios.
Option A: [tex]\(\sqrt{3}: \sqrt{3}\)[/tex]
The simplified ratio here is [tex]\(1:1\)[/tex]. This suggests that both legs are equal, which does not fit the ratio of the sides in a 30-60-90 triangle. Hence, option A is not correct.
Option B: [tex]\(1: \sqrt{3}\)[/tex]
This ratio perfectly matches the known ratio of the sides opposite the 30-degree and 60-degree angles in a 30-60-90 triangle. Hence, option B is correct.
Option C: [tex]\(1: \sqrt{2}\)[/tex]
This ratio does not match the ratio of the legs in a 30-60-90 triangle. This ratio corresponds to the sides of an isosceles right triangle (45-45-90 triangle) instead of a 30-60-90 triangle. Hence, option C is not correct.
Option D: [tex]\(\sqrt{2}: \sqrt{2}\)[/tex]
The simplified ratio here is again [tex]\(1:1\)[/tex]. Just like Option A, this suggests that both legs are equal, which is not characteristic of a 30-60-90 triangle. Hence, option D is not correct.
Option E: [tex]\(\sqrt{2}: \sqrt{3}\)[/tex]
To verify this, let's simplify [tex]\(\sqrt{2}: \sqrt{3}\)[/tex]:
1. Rewrite the ratio to compare with [tex]\(1: \sqrt{3}\)[/tex] of the legs:
[tex]\[ \frac{\sqrt{2}}{\sqrt{3}} = \frac{\sqrt{2} \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{\sqrt{6}}{3} \][/tex]
This ratio [tex]\(\frac{\sqrt{6}}{3}\)[/tex] is not equal to [tex]\(\frac{1}{\sqrt{3}}\)[/tex] or 1. Hence, option E is not correct.
Option F: [tex]\(\sqrt{3}: 3\)[/tex]
This can be simplified to:
[tex]\[ \frac{\sqrt{3}}{3} = \frac{\sqrt{3} \times \sqrt{3}}{3 \times \sqrt{3}} = \frac{3}{3 \sqrt{3}} = \frac{1}{\sqrt{3}} \][/tex]
This matches the ratio [tex]\(\frac{1}{\sqrt{3}}\)[/tex], though in a different form. Hence, it fits the ratio of the 30-60-90 triangle's legs. Therefore, option F is correct.
Thus, the correct options are:
B. [tex]\(1: \sqrt{3}\)[/tex] and F. [tex]\(\sqrt{3}: 3\)[/tex]
- Opposite the 30-degree angle: 1
- Opposite the 60-degree angle: [tex]\(\sqrt{3}\)[/tex]
- Opposite the 90-degree angle (the hypotenuse): 2
To determine which of the given options could be the ratio between the lengths of the two legs (the sides opposite the 30-degree and 60-degree angles), we use the known ratios.
Option A: [tex]\(\sqrt{3}: \sqrt{3}\)[/tex]
The simplified ratio here is [tex]\(1:1\)[/tex]. This suggests that both legs are equal, which does not fit the ratio of the sides in a 30-60-90 triangle. Hence, option A is not correct.
Option B: [tex]\(1: \sqrt{3}\)[/tex]
This ratio perfectly matches the known ratio of the sides opposite the 30-degree and 60-degree angles in a 30-60-90 triangle. Hence, option B is correct.
Option C: [tex]\(1: \sqrt{2}\)[/tex]
This ratio does not match the ratio of the legs in a 30-60-90 triangle. This ratio corresponds to the sides of an isosceles right triangle (45-45-90 triangle) instead of a 30-60-90 triangle. Hence, option C is not correct.
Option D: [tex]\(\sqrt{2}: \sqrt{2}\)[/tex]
The simplified ratio here is again [tex]\(1:1\)[/tex]. Just like Option A, this suggests that both legs are equal, which is not characteristic of a 30-60-90 triangle. Hence, option D is not correct.
Option E: [tex]\(\sqrt{2}: \sqrt{3}\)[/tex]
To verify this, let's simplify [tex]\(\sqrt{2}: \sqrt{3}\)[/tex]:
1. Rewrite the ratio to compare with [tex]\(1: \sqrt{3}\)[/tex] of the legs:
[tex]\[ \frac{\sqrt{2}}{\sqrt{3}} = \frac{\sqrt{2} \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{\sqrt{6}}{3} \][/tex]
This ratio [tex]\(\frac{\sqrt{6}}{3}\)[/tex] is not equal to [tex]\(\frac{1}{\sqrt{3}}\)[/tex] or 1. Hence, option E is not correct.
Option F: [tex]\(\sqrt{3}: 3\)[/tex]
This can be simplified to:
[tex]\[ \frac{\sqrt{3}}{3} = \frac{\sqrt{3} \times \sqrt{3}}{3 \times \sqrt{3}} = \frac{3}{3 \sqrt{3}} = \frac{1}{\sqrt{3}} \][/tex]
This matches the ratio [tex]\(\frac{1}{\sqrt{3}}\)[/tex], though in a different form. Hence, it fits the ratio of the 30-60-90 triangle's legs. Therefore, option F is correct.
Thus, the correct options are:
B. [tex]\(1: \sqrt{3}\)[/tex] and F. [tex]\(\sqrt{3}: 3\)[/tex]
