1. **Problem:** An airplane flies on a bearing of 120° for 200 km, then turns to a bearing of 210° for 150 km. Calculate its distance from the starting point. 2. **Formula and rules:** We use the Law of Cosines to find the distance between the starting point and the final position. The Law of Cosines states: $$c^2 = a^2 + b^2 - 2ab\cos(\theta)$$ where $a$ and $b$ are the lengths of two sides of a triangle, $\theta$ is the angle between them, and $c$ is the side opposite $\theta$. 3. **Step 1: Understand the bearings and angle between paths.** - The airplane first flies on a bearing of 120°. - Then it turns to a bearing of 210°. - The angle between these two bearings is $210^\circ - 120^\circ = 90^\circ$. 4. **Step 2: Apply the Law of Cosines.** - Let $a = 200$ km (first leg), $b = 150$ km (second leg), and $\theta = 90^\circ$. - Calculate $c$ (distance from start to final point): $$c^2 = 200^2 + 150^2 - 2 \times 200 \times 150 \times \cos(90^\circ)$$ - Since $\cos(90^\circ) = 0$, this simplifies to: $$c^2 = 40000 + 22500 = 62500$$ 5. **Step 3: Calculate $c$.** $$c = \sqrt{62500} = 250$$ 6. **Answer:** The airplane is 250 km from the starting point.