1. **Problem Statement:** Determine the shortest path from Start to Finish on the obstacle course given three possible routes: - A. Start to A to D to Finish - B. Start to B to C to D to Finish - C. Start to A to C to E to Finish 2. **Approach:** To find the shortest path, we need to consider the geometry of the paths and the angles given. The shortest path between two points is a straight line, but since the course has obstacles and angles, we use the Law of Cosines to calculate distances between points connected by angles. 3. **Law of Cosines:** For a triangle with sides $a$, $b$, and $c$, and angle $\theta$ opposite side $c$, the formula is: $$c^2 = a^2 + b^2 - 2ab\cos(\theta)$$ This helps us find unknown side lengths when two sides and the included angle are known. 4. **Calculate distances for each segment:** - From Start to A and Start to B, angles are given but no lengths, so assume unit lengths for simplicity (or equal lengths) since no lengths are provided. - Since the problem only provides angles and no lengths, and the paths are connected with given angles, the shortest path is the one with the least number of segments and smallest cumulative turning angles. 5. **Analyze each path:** - Path A: Start -> A -> D -> Finish - Path B: Start -> B -> C -> D -> Finish - Path C: Start -> A -> C -> E -> Finish 6. **Observations:** - Path A has 3 segments. - Path B has 4 segments. - Path C has 4 segments. 7. **Conclusion:** Since path A has fewer segments and the angles suggest less deviation, the shortest path is likely **A. Start to A to D to Finish**. **Final answer:** A. Start to A to D to Finish