Answered

Discover the answers you need at Westonci.ca, a dynamic Q&A platform where knowledge is shared freely by a community of experts. Explore thousands of questions and answers from knowledgeable experts in various fields on our Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.

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.
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.