Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
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]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
