Sure! Let's address the question step by step using the given results from the cosine calculation for [tex]$\triangle RST$[/tex].
Using the Law of Cosines, the equation for [tex]$\triangle RST$[/tex] is set up as follows:
[tex]\[ 5^2 = 7^2 + 3^2 - 2(7)(3) \cos(S) \][/tex]
Now, let's break down the steps to solve this equation systematically.
1. Calculate the left side of the equation:
[tex]\[ 5^2 = 25 \][/tex]
2. Calculate the right side of the equation without the cosine term:
[tex]\[ 7^2 + 3^2 = 49 + 9 = 58 \][/tex]
3. Find the remaining value after subtracting the left side from the right side (without cosine term):
[tex]\[ 25 - 58 = -33 \][/tex]
4. Calculate the denominator, which involves the cosine term:
[tex]\[ -2 \cdot 7 \cdot 3 = -42 \][/tex]
5. Finally, solve for [tex]\(\cos(S)\)[/tex] by dividing the remaining value by the denominator:
[tex]\[ \cos(S) = \frac{-33}{-42} = 0.7857142857142857 \][/tex]
Given that the calculations confirm the steps and the cosine value, we can now evaluate the list of given side lengths and combinations:
- [tex]\( r = 5 \text{ and } t = 7 \)[/tex]
- [tex]\( r = 3 \text{ and } t = 3 \)[/tex]
- [tex]\( s = 7 \text{ and } t = 5 \)[/tex]
- [tex]\( s = 5 \text{ and } t = 3 \)[/tex]
In our initial setup, we are told to use:
[tex]\[
a = 5, \quad b = 7, \quad c = 3
\][/tex]
Here:
- [tex]\(a = r = 5\)[/tex]
- [tex]\(b = s = 7\)[/tex]
- [tex]\(c = t = 3\)[/tex]
Matching these values with the given options, the correct side lengths match [tex]\( r = 5 \)[/tex] and [tex]\( t = 3 \)[/tex].
Therefore, the correct match from the options is:
[tex]\[ \boxed{s = 5 \text{ and } t = 3} \][/tex]
