Answered

Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Explore thousands of questions and answers from a knowledgeable community of experts ready to help you find solutions. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

The law of cosines for [tex]\triangle RST[/tex] can be set up as [tex]5^2 = 7^2 + 3^2 - 2(7)(3) \cos (S)[/tex]. What could be true about [tex]\triangle RST[/tex]?

Law of cosines: [tex]a^2 = b^2 + c^2 - 2bc \cos (A)[/tex]

A. [tex]r = 5[/tex] and [tex]t = 7[/tex]
B. [tex]r = 3[/tex] and [tex]t = 3[/tex]
C. [tex]s = 7[/tex] and [tex]t = 5[/tex]
D. [tex]s = 5[/tex] and [tex]t = 3[/tex]

Sagot :

GinnyAnswer
Alright, let's solve this step-by-step using the given parameters:

Given:
- [tex]\( r = 5 \)[/tex]
- [tex]\( s = 7 \)[/tex]
- [tex]\( t = 3 \)[/tex]

We want to verify if these values fit the given equation from the Law of Cosines:

[tex]\[ r^2 = s^2 + t^2 - 2 \cdot s \cdot t \cdot \cos(S) \][/tex]

Step-by-step solution:

1. Compute [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = 5^2 = 25 \][/tex]

2. Compute [tex]\( s^2 \)[/tex]:
[tex]\[ s^2 = 7^2 = 49 \][/tex]

3. Compute [tex]\( t^2 \)[/tex]:
[tex]\[ t^2 = 3^2 = 9 \][/tex]

4. Calculate [tex]\( 2 \cdot s \cdot t \)[/tex]:
[tex]\[ 2 \cdot s \cdot t = 2 \cdot 7 \cdot 3 = 42 \][/tex]

5. Substitute these values into the cosine equation:
[tex]\[ 25 = 49 + 9 - 42 \cdot \cos(S) \][/tex]
[tex]\[ 25 = 58 - 42 \cdot \cos(S) \][/tex]

6. Rearrange to isolate [tex]\(\cos(S)\)[/tex]:
[tex]\[ 25 - 58 = -42 \cdot \cos(S) \][/tex]
[tex]\[ -33 = -42 \cdot \cos(S) \][/tex]

7. Solve for [tex]\(\cos(S)\)[/tex]:
[tex]\[ \cos(S) = \frac{33}{42} \][/tex]
Simplifies to:
[tex]\[ \cos(S) = \frac{11}{14} \][/tex]

From this, we see that the provided values do indeed satisfy the Law of Cosines equation, and hence it confirms the triangle configuration. The relationship used confirms the sides [tex]\( r = 5 \)[/tex], [tex]\( s = 7 \)[/tex], and [tex]\( t = 3 \)[/tex].

Given the choices, the options that fit:
- [tex]\( r = 5 \)[/tex] and [tex]\( t = 7 \)[/tex]
- [tex]\( s = 7 \)[/tex] and [tex]\( t = 5 \)[/tex]
- [tex]\( s = 5 \)[/tex] and [tex]\( t = 3 \)[/tex]
- [tex]\( r = 3 \)[/tex] and [tex]\( t = 3 \)[/tex]

The correct values for [tex]\( r \)[/tex], [tex]\( s \)[/tex], and [tex]\( t \)[/tex] from the setup are [tex]\( r = 5 \)[/tex], [tex]\( s = 7 \)[/tex], and [tex]\( t = 3 \)[/tex]. Thus, none of the listed multiple-choice combinations perfectly align since they mix up [tex]\( r \)[/tex], [tex]\( s \)[/tex], and [tex]\( t \)[/tex]. Only the intersection of:
- [tex]\( r = 5\)[/tex], [tex]\( s = 7 \)[/tex], [tex]\( t = 3\)[/tex] match provided values fitting into the Law of Cosines equation configuration.
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.