1. **State the problem:** A surveyor needs to find the distance between points A and B, but a house obstructs the direct path. Given are side lengths $a=58$ feet, $b=75$ feet, and angle $C=83^\circ$. We need to find side $c$ to the nearest foot. 2. **Formula used:** We use the Law of Cosines, which relates the sides and angles of a triangle: $$c^2 = a^2 + b^2 - 2ab \cos C$$ This formula helps find the third side when two sides and the included angle are known. 3. **Substitute the known values:** $$c^2 = 58^2 + 75^2 - 2 \times 58 \times 75 \times \cos 83^\circ$$ 4. **Calculate each term:** $$58^2 = 3364$$ $$75^2 = 5625$$ $$2 \times 58 \times 75 = 8700$$ 5. **Calculate $\cos 83^\circ$:** $$\cos 83^\circ \approx 0.12187$$ 6. **Plug in the cosine value:** $$c^2 = 3364 + 5625 - 8700 \times 0.12187$$ 7. **Multiply:** $$8700 \times 0.12187 \approx 1060.17$$ 8. **Simplify:** $$c^2 = 3364 + 5625 - 1060.17 = 8989 - 1060.17 = 7928.83$$ 9. **Find $c$ by taking the square root:** $$c = \sqrt{7928.83} \approx 89.05$$ 10. **Round to the nearest foot:** $$c \approx 89$$ feet **Final answer:** The distance between points A and B is approximately **89 feet**.