1. **Problem statement:** A boat sails from point A due north for 7 km to point B. From B, it sails on a bearing of 100° for 10 km to point C. We need to find the distance from point C back to point A. 2. **Understanding bearings and directions:** Bearing is measured clockwise from the north direction. A bearing of 100° means the boat moves 10° south of east from point B. 3. **Set up coordinate system:** Place point A at the origin $(0,0)$. - Since the boat sails due north 7 km to B, coordinates of B are $(0,7)$. 4. **Find coordinates of C:** From B, the boat moves 10 km at bearing 100°. - The angle from the east axis is $100° - 90° = 10°$ south of east. - So, the displacement vector from B to C is: $$x = 10 \times \cos(10^\circ)$$ $$y = -10 \times \sin(10^\circ)$$ - Calculate: $$x \approx 10 \times 0.9848 = 9.848$$ $$y \approx -10 \times 0.1736 = -1.736$$ - Coordinates of C: $$C_x = B_x + x = 0 + 9.848 = 9.848$$ $$C_y = B_y + y = 7 - 1.736 = 5.264$$ 5. **Calculate distance from A to C:** Use distance formula: $$d = \sqrt{(C_x - 0)^2 + (C_y - 0)^2} = \sqrt{9.848^2 + 5.264^2}$$ $$= \sqrt{96.97 + 27.71} = \sqrt{124.68} \approx 11.17$$ **Final answer:** The distance from point C to point A is approximately **11.17 km**.