Question

on which triangle can the law of cosines be used to find the length of an unknown side?
law of cosines: $a^2 = b^2 + c^2 - 2bc\cos(a)$

Explanation:

Step1: Recall Law of Cosines use case

The Law of Cosines $a^2 = b^2 + c^2 - 2bc\cos(A)$ is used to find an unknown side when two sides and the included angle (SAS) are known, or all three sides to find an angle.

Step2: Analyze each triangle

1. First triangle: All angles and one side are known (AAS/ASA), so Law of Sines is used instead.
2. Second triangle: Two sides ($RS=7$, $QS=12$) and their included angle ($\angle S=57^\circ$) are known. This fits the SAS condition for Law of Cosines to find unknown side $s$.
3. Third triangle: Only two sides are known, no included angle provided, so Law of Cosines cannot be applied.

Answer:

The second triangle (with sides 12, 7, included angle $57^\circ$ at vertex S)