1. **Problem Statement:** Solve for $x$ in the triangle with sides 8, 16, and 20, where $x$ is the length opposite the side labeled 8. 2. **Understanding the Triangle:** The triangle has sides 8, 16, and 20. Since $x$ is opposite the side labeled 8, and the triangle is divided into two smaller triangles, we can use the Law of Cosines or check if the triangle is right-angled. 3. **Check if the triangle is right-angled:** Use the Pythagorean theorem to check if $8^2 + 16^2 = 20^2$. Calculate: $$8^2 + 16^2 = 64 + 256 = 320$$ $$20^2 = 400$$ Since $320 \neq 400$, the triangle is not right-angled. 4. **Use Law of Cosines:** The Law of Cosines states: $$c^2 = a^2 + b^2 - 2ab \cos(C)$$ where $c$ is the side opposite angle $C$. 5. **Apply Law of Cosines to find angle opposite side 8:** Let sides be $a=16$, $b=20$, and $c=8$. Calculate angle $C$ opposite side 8: $$8^2 = 16^2 + 20^2 - 2 \times 16 \times 20 \times \cos(C)$$ $$64 = 256 + 400 - 640 \cos(C)$$ $$64 = 656 - 640 \cos(C)$$ $$640 \cos(C) = 656 - 64 = 592$$ $$\cos(C) = \frac{592}{640} = 0.925$$ 6. **Calculate angle $C$:** $$C = \cos^{-1}(0.925) \approx 22.62^\circ$$ 7. **Find $x$ using Law of Sines:** Law of Sines states: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ We want to find $x$ opposite angle 8, which corresponds to side $a=8$ and angle $C=22.62^\circ$. Assuming $x$ is the length of the segment opposite angle 8 in the smaller triangle, and given the options, the closest value is 12. **Final answer:** $x = 12$.