1. **State the problem:** We are given a triangle $\triangle DEF$ with sides $f = 830$ cm, $d = 520$ cm, and angle $\angle E = 129^\circ$. We need to find the length of side $e$ opposite angle $E$. 2. **Formula used:** The Law of Cosines states: $$e^2 = d^2 + f^2 - 2df \cos(E)$$ This formula relates the lengths of sides of a triangle to the cosine of one of its angles. 3. **Substitute the known values:** $$e^2 = 520^2 + 830^2 - 2 \times 520 \times 830 \times \cos(129^\circ)$$ 4. **Calculate each term:** $$520^2 = 270400$$ $$830^2 = 688900$$ 5. **Calculate the cosine:** $$\cos(129^\circ) \approx -0.6561$$ 6. **Calculate the product:** $$2 \times 520 \times 830 = 863200$$ 7. **Calculate the entire expression:** $$e^2 = 270400 + 688900 - 863200 \times (-0.6561)$$ $$e^2 = 959300 + 566525.52$$ $$e^2 = 1,525,825.52$$ 8. **Find $e$ by taking the square root:** $$e = \sqrt{1,525,825.52} \approx 1235.27$$ 9. **Final answer:** The length of side $e$ is approximately $1235$ cm to the nearest centimeter.