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]
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]
