Which of the following could be the ratio of the length of the longer leg of a [tex]30^\circ-60^\circ-90^\circ[/tex] triangle to the length of its hypotenuse?

Check all that apply.

A. [tex]2 : 3 \sqrt{3}[/tex]
B. [tex]1 : \sqrt{2}[/tex]
C. [tex]2 : 2 \sqrt{2}[/tex]
D. [tex]\sqrt{3} : 2[/tex]
E. [tex]3 : 2 \sqrt{3}[/tex]
F. [tex]\sqrt{2} : \sqrt{3}[/tex]



Answer :

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]