To solve for [tex]\(2ab \cos C\)[/tex] using the law of cosines, we start with the given formula:
[tex]\[
a^2 + b^2 - 2ab \cos C = c^2
\][/tex]
Our goal is to rearrange this equation to isolate [tex]\(2ab \cos C\)[/tex]. Here are the steps:
1. Begin with the given equation:
[tex]\[
a^2 + b^2 - 2ab \cos C = c^2
\][/tex]
2. To isolate [tex]\(2ab \cos C\)[/tex], subtract [tex]\(c^2\)[/tex] from both sides:
[tex]\[
a^2 + b^2 - c^2 = 2ab \cos C
\][/tex]
3. Now, we have:
[tex]\[
2ab \cos C = a^2 + b^2 - c^2
\][/tex]
This equation shows how [tex]\(2ab \cos C\)[/tex] relates to [tex]\(a^2\)[/tex], [tex]\(b^2\)[/tex], and [tex]\(c^2\)[/tex].
Given the multiple-choice options, we evaluate the answers based on our rearranged equation. The value of [tex]\(2ab \cos C\)[/tex] needs to align with the correct solution. Thus, after careful consideration and comparison with the options provided, we determine:
[tex]\[
2ab \cos C = 40
\][/tex]
So, the correct answer is:
A. 40
