1. **Problem statement:** Given a triangle with an angle of $23^\circ$ and two sides measuring 22.3 cm (opposite the $23^\circ$ angle) and 16.5 cm, find the length of the third side $x$. 2. **Formula used:** We can use the Law of Cosines to find the unknown side $x$. The Law of Cosines states: $$x^2 = a^2 + b^2 - 2ab \cos(C)$$ where $a$ and $b$ are the known sides, and $C$ is the included angle between them. 3. **Identify sides and angle:** Here, $a = 16.5$ cm, $b = 22.3$ cm, and $C = 23^\circ$. 4. **Apply the formula:** $$x^2 = 16.5^2 + 22.3^2 - 2 \times 16.5 \times 22.3 \times \cos(23^\circ)$$ 5. **Calculate each term:** $$16.5^2 = 272.25$$ $$22.3^2 = 497.29$$ $$2 \times 16.5 \times 22.3 = 735.9$$ $$\cos(23^\circ) \approx 0.9205$$ 6. **Substitute values:** $$x^2 = 272.25 + 497.29 - 735.9 \times 0.9205$$ $$x^2 = 769.54 - 677.56 = 91.98$$ 7. **Find $x$ by taking the square root:** $$x = \sqrt{91.98} \approx 9.59$$ **Final answer:** The length of side $x$ is approximately $9.59$ cm.