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.
