1. **Problem Statement:** We have two bearings from the north direction: 135° and 060°, with distances 8 km and 15 km respectively. We want to find the relative position of the two points described by these bearings and distances. 2. **Understanding Bearings:** Bearings are measured clockwise from the north direction. - Bearing 135° means the point is located southeast. - Bearing 060° means the point is located northeast. 3. **Convert Bearings to Cartesian Coordinates:** We use the formulas: $$x = d \times \sin(\theta)$$ $$y = d \times \cos(\theta)$$ where $d$ is the distance and $\theta$ is the bearing angle. 4. **Calculate Coordinates for Each Point:** - For bearing 135° and distance 8 km: $$x_1 = 8 \times \sin(135^\circ) = 8 \times \frac{\sqrt{2}}{2} = 8 \times 0.7071 = 5.657$$ $$y_1 = 8 \times \cos(135^\circ) = 8 \times -\frac{\sqrt{2}}{2} = 8 \times -0.7071 = -5.657$$ - For bearing 060° and distance 15 km: $$x_2 = 15 \times \sin(60^\circ) = 15 \times \frac{\sqrt{3}}{2} = 15 \times 0.8660 = 12.990$$ $$y_2 = 15 \times \cos(60^\circ) = 15 \times \frac{1}{2} = 7.5$$ 5. **Interpretation:** - Point 1 is at approximately $(5.657, -5.657)$ km. - Point 2 is at approximately $(12.990, 7.5)$ km. 6. **Summary:** Using the bearings and distances, we converted polar coordinates to Cartesian coordinates to find the relative positions of the two points from the origin (north direction). **Final coordinates:** - Point 1: $(5.657, -5.657)$ km - Point 2: $(12.990, 7.5)$ km