Let us examine the given options using the law of cosines. The law of cosines states that for any triangle with sides \(a\), \(b\), and \(c\) opposite to angles \(A\), \(B\), and \(C\) respectively, the following formula holds:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Given the triangle sides and angle, we can identify the values as follows:
- \( a = 9 \)
- \( b = 10 \)
- \( C = 100^\circ \)
Now, let's verify each provided equation:
1. \( 9^2 = s^2 + 10^2 - 2(s)(10) \cos(100^\circ) \)
Rewriting this equation, we get:
[tex]\[ 81 = s^2 + 100 - 20s \cos(100^\circ) \][/tex]
This does not match the standard form of the law of cosines (\(s\) should be squared and isolated on one side of the equation), so this is incorrect.
2. \( 9 = s + 10 - 2(s)(10) \cos(100^\circ) \)
This equation is not correct because it does not square the sides or follow the law of cosines structure at all.
3. \( 10^2 = s^2 + 100 - 2(s)(10) \cos(100^\circ) \)
Rewriting this equation, we get:
[tex]\[ 100 = s^2 + 100 - 20s \cos(100^\circ) \][/tex]
Again, this does not follow the correct pattern as expected by the law of cosines since \(s\) is not appropriately isolated.
4. \( s^2 = 9^2 + 10^2 - 2(9)(10) \cos(100^\circ) \)
Rewriting this equation, we get:
[tex]\[ s^2 = 81 + 100 - 2 \times 9 \times 10 \times \cos(100^\circ) \][/tex]
[tex]\[ s^2 = 181 - 180 \cos(100^\circ) \][/tex]
This correctly fits the law of cosines form, where the terms are correctly placed and simplified.
Thus, option 4:
[tex]\[ s^2 = 9^2 + 10^2 - 2(9)(10) \cos(100^\circ) \][/tex]
is the correctly applied law of cosines to solve for the length [tex]\( s \)[/tex].
