The law of cosines states for a triangle ABC with side lengths a, b, and 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 Law of Cosines has three forms, one for each side of the triangle. The general form is $c^2 = a^2 + b^2 - 2ab \cos(C)$, where $c$ is the side opposite angle $C$, and $a$ and $b$ are the other two sides. Similarly, $a^2 = b^2 + c^2 - 2bc \cos(A)$ and $b^2 = a^2 + c^2 - 2ac \cos(B)$.

3. Match the Formula to the Options: We need to find the option that correctly states the Law of Cosines. The image shows a triangle ABC with sides labeled $a$, $b$, and $c$ opposite to vertices A, B, and C respectively. The question is asking for the law of cosines statement. Let's examine the given options:
A. $c^2 = a^2 - b^2 - 2bccos(C)$ - This is incorrect because it has a minus sign before $b^2$ and uses $bc$ instead of $ab$ with $\cos(C)$.
B. $c^2 = a^2 + b^2 - 2abcos(C)$ - This matches the standard form of the Law of Cosines when solving for side $c$ using angle $C$.
C. $c^2 = a^2 + b^2 - 2bc cos(B)$ - This is incorrect because it uses $\cos(B)$ instead of $\cos(C)$ and the product of sides $bc$ instead of $ab$.
D. $c^2 = a^2 + b^2 - 2bc cos(A)$ - This is incorrect because it uses $\cos(A)$ instead of $\cos(C)$ and the product of sides $bc$ instead of $ab$.

1. Select the Correct Option: Based on the comparison, option B is the correct statement of the Law of Cosines for side $c$.