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

Check all that apply.

A. [tex]\sqrt{2}: \sqrt{3}[/tex]
B. [tex]2: 2 \sqrt{2}[/tex]
C. [tex]3: 2 \sqrt{3}[/tex]
D. [tex]\sqrt{3}: 2[/tex]
E. [tex]2: 3 \sqrt{5}[/tex]
F. [tex]1: \sqrt{2}[/tex]



Answer :

To solve the problem of determining which of the given ratios could be the ratio of the longer leg of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle to its hypotenuse, we need to review the properties of this special right triangle.

In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, the sides are in a specific ratio. The side opposite the [tex]\(30^\circ\)[/tex] angle (shorter leg) is [tex]\(a\)[/tex], the side opposite the [tex]\(60^\circ\)[/tex] angle (longer leg) is [tex]\(a\sqrt{3}\)[/tex], and the hypotenuse is [tex]\(2a\)[/tex]. Therefore, the ratio of the longer leg to the hypotenuse is:

[tex]\[ \frac{a\sqrt{3}}{2a} = \frac{\sqrt{3}}{2} \][/tex]

Now let's examine each given ratio to see if it simplifies to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].

### Option A: [tex]\(\frac{\sqrt{2}}{\sqrt{3}}\)[/tex]

[tex]\[ \frac{\sqrt{2}}{\sqrt{3}} \neq \frac{\sqrt{3}}{2} \][/tex]

So, option A is not valid.

### Option B: [tex]\(\frac{2}{2\sqrt{2}}\)[/tex]

Simplify the ratio by dividing both the numerator and the denominator by 2:

[tex]\[ \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} \][/tex]

[tex]\[ \frac{1}{\sqrt{2}} \neq \frac{\sqrt{3}}{2} \][/tex]

So, option B is not valid.

### Option C: [tex]\(\frac{3}{2\sqrt{3}}\)[/tex]

Simplify the ratio by dividing both the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]:

[tex]\[ \frac{3}{2\sqrt{3}} = \frac{3}{2\sqrt{3}} \cdot \frac{1}{\sqrt{3}} = \frac{3}{2 \cdot 3} = \frac{1}{2} \][/tex]

[tex]\[ \frac{3}{2\sqrt{3}} \neq \frac{\sqrt{3}}{2} \][/tex]

So, option C is not valid.

### Option D: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]

This is exactly the correct ratio that we are looking for:

[tex]\[ \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \][/tex]

So, option D is valid.

### Option E: [tex]\(\frac{2}{3\sqrt{5}}\)[/tex]

Simplify the ratio by dividing both the numerator and the denominator by 2:

[tex]\[ \frac{2}{3\sqrt{5}} \neq \frac{\sqrt{3}}{2} \][/tex]

So, option E is not valid.

### Option F: [tex]\(\frac{1}{\sqrt{2}}\)[/tex]

Simplify the expression:

[tex]\[ \frac{1}{\sqrt{2}} \ne \frac{\sqrt{3}}{2} \][/tex]

So, option F is not valid.

Therefore, the only valid ratio for the length of the longer leg of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle to the length of its hypotenuse is:

D. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]