Answered

Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Connect with professionals on our platform to receive accurate answers to your questions quickly and efficiently. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

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]3 \sqrt{3} : 6[/tex]
B. [tex]\sqrt{3} : \sqrt{3}[/tex]
C. [tex]\sqrt{3} : 2[/tex]
D. [tex]3 : 2 \sqrt{3}[/tex]
E. [tex]1 : \sqrt{3}[/tex]
F. [tex]\sqrt{2} : \sqrt{3}[/tex]

Sagot :

GinnyAnswer
A 30-60-90 triangle has specific side ratios based on its angles. The sides of a 30-60-90 triangle have the ratio:

1. The shortest leg (opposite the 30° angle) is [tex]\( \frac{1}{2} \)[/tex] the hypotenuse.
2. The longer leg (opposite the 60° angle) is [tex]\( \frac{\sqrt{3}}{2} \)[/tex] times the hypotenuse.
3. The hypotenuse is the longest side.

Given the properties of a 30-60-90 triangle, let's determine the ratio of the longer leg to the hypotenuse.

The ratio of the longer leg to the hypotenuse in a 30-60-90 triangle is:
[tex]\[ \frac{\frac{\sqrt{3}}{2}}{1} = \frac{\sqrt{3}}{2}. \][/tex]

We need to examine each given option to see if it simplifies to [tex]\( \frac{\sqrt{3}}{2} \)[/tex].

### Option A: [tex]\( \frac{3\sqrt{3}}{6} \)[/tex]
Simplify:
[tex]\[ \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2}. \][/tex]
This matches [tex]\( \frac{\sqrt{3}}{2} \)[/tex].

### Option B: [tex]\( \sqrt{3}:\sqrt{3} \)[/tex]
Simplify:
[tex]\[ \frac{\sqrt{3}}{\sqrt{3}} = 1. \][/tex]
This does not match [tex]\( \frac{\sqrt{3}}{2} \)[/tex].

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

### Option D: [tex]\( \frac{3}{2\sqrt{3}} \)[/tex]
Simplify:
[tex]\[ \frac{3}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2}. \][/tex]
This matches [tex]\( \frac{\sqrt{3}}{2} \)[/tex].

### Option E: [tex]\( \frac{1}{\sqrt{3}} \)[/tex]
Simplify:
[tex]\[ \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}. \][/tex]
This does not match [tex]\( \frac{\sqrt{3}}{2} \)[/tex].

### Option F: [tex]\( \frac{\sqrt{2}}{\sqrt{3}} \)[/tex]
Simplify:
[tex]\[ \frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{6}}{3}. \][/tex]
This does not match [tex]\( \frac{\sqrt{3}}{2} \)[/tex].

In conclusion, the options that could be the ratio of the length of the longer leg of a 30-60-90 triangle to the length of its hypotenuse are:
- A: [tex]\( \frac{3\sqrt{3}}{6} \)[/tex]
- C: [tex]\( \frac{\sqrt{3}}{2} \)[/tex]
- D: [tex]\( \frac{3}{2\sqrt{3}} \)[/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.