Law of cosines states for a triangle ABC with side lengths a, b, c

1. Identify the Law of Cosines: The problem asks to identify the correct statement of the Law of Cosines for a triangle ABC with side lengths a, b, and c. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.

1. Recall the Law of Cosines Formula: The general form of the Law of Cosines can be expressed in three ways, depending on which side is being solved for. If we want to find side $c$, the formula is: $$c^2 = a^2 + b^2 - 2ab\cos(C)$$ where $C$ is the angle opposite side $c$.

3. Analyze the Options: Let's examine the given options:
A. $c^2 = a^2 - b^2 - 2bccos(C)$: This formula is incorrect. It subtracts $b^2$ instead of adding it, and the term $2bccos(C)$ is also incorrect in this form.
B. $c^2 = a^2 + b^2 - 2abcos(C)$: This matches the standard form of the Law of Cosines for finding side $c$, where $C$ is the angle opposite side $c$.
C. $c^2 = a^2 + b^2 - 2bccos(B)$: This formula is incorrect. The angle used should be $C$, not $B$, when solving for $c^2$ using sides $a$ and $b$.
D. $c^2 = a^2 + b^2 - 2bccos(A)$: This formula is incorrect. The angle used should be $C$, not $A$, when solving for $c^2$ using sides $a$ and $b$. Also, the term $2bccos(A)$ is not directly related to finding $c^2$ in this manner.

1. Select the Correct Option: Based on the analysis, option B correctly states the Law of Cosines for finding the square of side $c$.