1. **Problem Statement:** Determine which formula correctly represents the Law of Cosines for triangle $\triangle XYZ$. 2. **Recall the Law of Cosines formula:** For any triangle with sides $a, b, c$ opposite angles $A, B, C$ respectively, the Law of Cosines states: $$a^2 = b^2 + c^2 - 2bc \cos A$$ This formula relates the lengths of the sides to the cosine of the included angle. 3. **Apply to $\triangle XYZ$:** Assuming side $x$ is opposite angle $X$, and sides $y$ and $z$ are the other two sides, the formula becomes: $$x^2 = y^2 + z^2 - 2yz \cos X$$ 4. **Check each option:** - a. $x^2 = y^2 + z^2 - 2xy \cos X$ — Incorrect because the last term should be $2yz$, not $2xy$. - b. $x^2 = y^2 + z^2 - 2zy \cos X$ — Correct, since multiplication is commutative, $2zy = 2yz$. - c. $x^2 = y^2 + z^2 - 2yz \sin X$ — Incorrect, the Law of Cosines uses cosine, not sine. - d. $y^2 = x^2 + z^2 - 2xz \cos X$ — Incorrect, the side opposite angle $X$ is $x$, so this formula is mismatched. 5. **Conclusion:** The correct formula is option b: $$x^2 = y^2 + z^2 - 2zy \cos X$$ This formula allows you to find the length of side $x$ when you know sides $y$, $z$, and the included angle $X$.