1. **Stating the problem:** We have a triangle with two sides measuring 126 m and 100 m, and the included angle between these sides is 75°. We want to find the length of the third side (the path). 2. **Formula used:** To find the length of the third side in a triangle when two sides and the included angle are known, we use the Law of Cosines: $$c^2 = a^2 + b^2 - 2ab \cos(C)$$ where $a = 126$, $b = 100$, and $C = 75^\circ$. 3. **Applying the formula:** $$c^2 = 126^2 + 100^2 - 2 \times 126 \times 100 \times \cos(75^\circ)$$ 4. **Calculate each term:** $$126^2 = 15876$$ $$100^2 = 10000$$ $$2 \times 126 \times 100 = 25200$$ 5. **Calculate $\cos(75^\circ)$:** $$\cos(75^\circ) \approx 0.2588$$ 6. **Substitute values:** $$c^2 = 15876 + 10000 - 25200 \times 0.2588$$ $$c^2 = 25876 - 6521.76$$ $$c^2 = 19354.24$$ 7. **Find $c$ by taking the square root:** $$c = \sqrt{19354.24} \approx 139.15$$ 8. **Answer:** The length of the path is approximately **139.15 meters**.