To determine which equation correctly uses the law of cosines to solve for the length [tex]\( s \)[/tex], we need to start by recalling the law of cosines formula. For a triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] opposite the respective angles [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex], the law of cosines states:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cos C
\][/tex]
Given the problem, let's identify the variables:
- [tex]\( a = 9 \)[/tex]
- [tex]\( b = 10 \)[/tex]
- [tex]\( C = 100^\circ \)[/tex]
- [tex]\( s \)[/tex] is the side opposite the given angle [tex]\( 100^\circ \)[/tex]
Using the law of cosines, the correct equation to solve for [tex]\( s \)[/tex] would be:
[tex]\[
s^2 = 9^2 + 10^2 - 2(9)(10) \cos(100^\circ)
\][/tex]
We can now compare this equation to the provided options:
1. [tex]\( 9^2 = s^2 + 10^2 - 2(s)(10) \cos(100^\circ) \)[/tex]
2. [tex]\( 9 = s + 10 - 2(s)(10) \cos(100^\circ) \)[/tex]
3. [tex]\( 10^2 = s^2 + 100 - 2(s)(10) \cos(100^\circ) \)[/tex]
4. [tex]\( s^2 = 9^2 + 10^2 - 2(9)(10) \cos(100^\circ) \)[/tex]
Comparing each option to the equation derived from the law of cosines, we see that option 4 matches exactly. Therefore, the correct equation that uses the law of cosines to solve for the length [tex]\( s \)[/tex] is:
[tex]\[
s^2 = 9^2 + 10^2 - 2(9)(10) \cos(100^\circ)
\][/tex]
