1. **Stating the problem:** We are given a triangle with sides 7 cm, 9 cm, and 12 cm, and we need to find the size of the angle opposite the 7 cm side. 2. **Formula used:** To find an angle in a triangle when all three sides are known, we use the cosine rule: $$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$ where $a$ is the side opposite the angle $A$, and $b$ and $c$ are the other two sides. 3. **Assigning values:** Let $a = 7$, $b = 9$, and $c = 12$. 4. **Calculate the cosine of the angle:** $$\cos A = \frac{9^2 + 12^2 - 7^2}{2 \times 9 \times 12} = \frac{81 + 144 - 49}{216} = \frac{176}{216}$$ 5. **Simplify the fraction:** $$\frac{176}{216} = \frac{\cancel{16} \times 11}{\cancel{16} \times 13.5} = \frac{11}{13.5}$$ 6. **Calculate the decimal value:** $$\cos A \approx 0.8148$$ 7. **Find the angle $A$ by taking the inverse cosine:** $$A = \cos^{-1}(0.8148) \approx 35.7^\circ$$ **Final answer:** The angle opposite the 7 cm side is approximately **35.7 degrees**.