Answered

Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our 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.
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.