1. **Problem statement:** Calculate the length of side $K$ opposite the $14^\circ$ angle in a triangle with sides 9 cm and 14 cm. 2. **Formula:** Use the Cosine Rule: $$a^2 = b^2 + c^2 - 2bc \cos A$$ where $a$ is the side opposite angle $A$. 3. **Apply values:** Here, $a = K$, $b = 9$, $c = 14$, and $A = 14^\circ$. 4. **Calculation:** $$K^2 = 9^2 + 14^2 - 2 \times 9 \times 14 \times \cos 14^\circ$$ $$K^2 = 81 + 196 - 252 \times \cos 14^\circ$$ 5. **Evaluate cosine:** $$\cos 14^\circ \approx 0.9703$$ 6. **Substitute:** $$K^2 = 277 - 252 \times 0.9703 = 277 - 244.5 = 32.5$$ 7. **Find $K$:** $$K = \sqrt{32.5} \approx 5.7 \text{ cm}$$ **Final answer:** The length of side $K$ is approximately 5.7 cm.