Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

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

Sagot :

To determine which of the given ratios 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, we need to recall that in a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, the ratio of the sides opposite these angles is as follows:

- The side opposite the [tex]\(30^\circ\)[/tex] angle (the shorter leg) is [tex]\(a\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle (the longer leg) is [tex]\(\sqrt{3}a\)[/tex].
- The hypotenuse is [tex]\(2a\)[/tex].

Therefore, the ratio of the longer leg to the hypotenuse is:
[tex]\[ \frac{\text{Longer leg}}{\text{Hypotenuse}} = \frac{\sqrt{3}a}{2a} = \frac{\sqrt{3}}{2} \][/tex]

Now let's analyze each given option to see which ratios match [tex]\(\frac{\sqrt{3}}{2}\)[/tex].

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

[tex]\[ \frac{2}{3\sqrt{3}} = \frac{2}{3\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{9} \][/tex]
This is not equal to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].

Option B: [tex]\(1: \sqrt{2}\)[/tex]

[tex]\[ \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \][/tex]
This is not equal to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].

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

[tex]\[ \frac{2}{2\sqrt{2}} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \][/tex]
This is not equal to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].

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

[tex]\[ \frac{\sqrt{3}}{2} \][/tex]
This matches exactly [tex]\(\frac{\sqrt{3}}{2}\)[/tex].

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

[tex]\[ \frac{3}{2\sqrt{3}} = \frac{3}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2} \][/tex]
This matches exactly [tex]\(\frac{\sqrt{3}}{2}\)[/tex].

Option F: [tex]\(\sqrt{2}: \sqrt{3}\)[/tex]

[tex]\[ \frac{\sqrt{2}}{\sqrt{3}} = \sqrt{\frac{2}{3}} \][/tex]
This is not equal to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].

Therefore, the ratios that match [tex]\(\frac{\sqrt{3}}{2}\)[/tex] are Options D and E.

So, the correct options are:
- D. [tex]\(\sqrt{3}: 2\)[/tex]
- E. [tex]\(3: 2\sqrt{3}\)[/tex]