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To determine which of the given ratios could represent the ratio of the longer leg of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle to its hypotenuse, we first need to understand the properties of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle:
1. In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle:
- The shorter leg (opposite the [tex]\(30^\circ\)[/tex] angle) has a length [tex]\(x\)[/tex].
- The longer leg (opposite the [tex]\(60^\circ\)[/tex] angle) has a length [tex]\(x\sqrt{3}\)[/tex].
- The hypotenuse (opposite the [tex]\(90^\circ\)[/tex] angle) has a length [tex]\(2x\)[/tex].
2. The ratio of the longer leg to the hypotenuse in such a triangle is:
[tex]\[ \frac{\text{longer leg}}{\text{hypotenuse}} = \frac{x\sqrt{3}}{2x} = \frac{\sqrt{3}}{2} \][/tex]
Let's check each option to see if it matches [tex]\(\frac{\sqrt{3}}{2}\)[/tex]:
Option A: [tex]\(2 : 3\sqrt{3}\)[/tex]
- Converting to fraction form:
[tex]\[ \frac{2}{3\sqrt{3}} \][/tex]
- Simplifying:
[tex]\[ \frac{2}{3\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{9} \][/tex]
[tex]\(\frac{2\sqrt{3}}{9} \neq \frac{\sqrt{3}}{2}\)[/tex]
Option A is not correct.
Option B: [tex]\(1 : \sqrt{2}\)[/tex]
- Converting to fraction form:
[tex]\[ \frac{1}{\sqrt{2}} \][/tex]
[tex]\(\frac{1}{\sqrt{2}} \neq \frac{\sqrt{3}}{2}\)[/tex]
Option B is not correct.
Option C: [tex]\(2 : 2\sqrt{2}\)[/tex]
- Converting to fraction form:
[tex]\[ \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} \][/tex]
[tex]\(\frac{1}{\sqrt{2}} \neq \frac{\sqrt{3}}{2}\)[/tex]
Option C is not correct.
Option D: [tex]\(\sqrt{3} : 2\)[/tex]
- Converting to fraction form:
[tex]\[ \frac{\sqrt{3}}{2} \][/tex]
[tex]\(\frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}\)[/tex]
Option D is correct.
Option E: [tex]\(3 : 2\sqrt{3}\)[/tex]
- Converting to fraction form:
[tex]\[ \frac{3}{2\sqrt{3}} \][/tex]
- Simplifying:
[tex]\[ \frac{3}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2} \][/tex]
Option E is correct.
Option F: [tex]\(\sqrt{2} : \sqrt{3}\)[/tex]
- Converting to fraction form:
[tex]\[ \frac{\sqrt{2}}{\sqrt{3}} \][/tex]
[tex]\(\frac{\sqrt{2}}{\sqrt{3}} \neq \frac{\sqrt{3}}{2}\)[/tex]
Option F is not correct.
Therefore, the correct answers are:
- Option D: [tex]\(\sqrt{3} : 2\)[/tex]
- Option E: [tex]\(3 : 2\sqrt{3}\)[/tex]
1. In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle:
- The shorter leg (opposite the [tex]\(30^\circ\)[/tex] angle) has a length [tex]\(x\)[/tex].
- The longer leg (opposite the [tex]\(60^\circ\)[/tex] angle) has a length [tex]\(x\sqrt{3}\)[/tex].
- The hypotenuse (opposite the [tex]\(90^\circ\)[/tex] angle) has a length [tex]\(2x\)[/tex].
2. The ratio of the longer leg to the hypotenuse in such a triangle is:
[tex]\[ \frac{\text{longer leg}}{\text{hypotenuse}} = \frac{x\sqrt{3}}{2x} = \frac{\sqrt{3}}{2} \][/tex]
Let's check each option to see if it matches [tex]\(\frac{\sqrt{3}}{2}\)[/tex]:
Option A: [tex]\(2 : 3\sqrt{3}\)[/tex]
- Converting to fraction form:
[tex]\[ \frac{2}{3\sqrt{3}} \][/tex]
- Simplifying:
[tex]\[ \frac{2}{3\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{9} \][/tex]
[tex]\(\frac{2\sqrt{3}}{9} \neq \frac{\sqrt{3}}{2}\)[/tex]
Option A is not correct.
Option B: [tex]\(1 : \sqrt{2}\)[/tex]
- Converting to fraction form:
[tex]\[ \frac{1}{\sqrt{2}} \][/tex]
[tex]\(\frac{1}{\sqrt{2}} \neq \frac{\sqrt{3}}{2}\)[/tex]
Option B is not correct.
Option C: [tex]\(2 : 2\sqrt{2}\)[/tex]
- Converting to fraction form:
[tex]\[ \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} \][/tex]
[tex]\(\frac{1}{\sqrt{2}} \neq \frac{\sqrt{3}}{2}\)[/tex]
Option C is not correct.
Option D: [tex]\(\sqrt{3} : 2\)[/tex]
- Converting to fraction form:
[tex]\[ \frac{\sqrt{3}}{2} \][/tex]
[tex]\(\frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}\)[/tex]
Option D is correct.
Option E: [tex]\(3 : 2\sqrt{3}\)[/tex]
- Converting to fraction form:
[tex]\[ \frac{3}{2\sqrt{3}} \][/tex]
- Simplifying:
[tex]\[ \frac{3}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2} \][/tex]
Option E is correct.
Option F: [tex]\(\sqrt{2} : \sqrt{3}\)[/tex]
- Converting to fraction form:
[tex]\[ \frac{\sqrt{2}}{\sqrt{3}} \][/tex]
[tex]\(\frac{\sqrt{2}}{\sqrt{3}} \neq \frac{\sqrt{3}}{2}\)[/tex]
Option F is not correct.
Therefore, the correct answers are:
- Option D: [tex]\(\sqrt{3} : 2\)[/tex]
- Option E: [tex]\(3 : 2\sqrt{3}\)[/tex]
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