1. **Problem Statement:** From a 14 m high tower, the angles of depression to two houses are 7° and 4°. We need to find the distance between the houses under three scenarios: a) Houses on the same side of the tower. b) Houses on opposite sides of the tower. c) One house due South and the other due East from the tower. 2. **Formulas and Rules:** - The height of the tower is the vertical leg of right triangles formed with the houses. - The horizontal distances to the houses can be found using the tangent of the angle of depression: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{14}{d} \implies d = \frac{14}{\tan(\theta)}$$ - Angles of depression equal angles of elevation from the houses to the top of the tower. 3. **Calculations:** - Distance to house 1 (angle 7°): $$d_1 = \frac{14}{\tan(7^\circ)}$$ - Distance to house 2 (angle 4°): $$d_2 = \frac{14}{\tan(4^\circ)}$$ Calculate values: $$\tan(7^\circ) \approx 0.12278, \quad d_1 = \frac{14}{0.12278} \approx 114.01\,m$$ $$\tan(4^\circ) \approx 0.06993, \quad d_2 = \frac{14}{0.06993} \approx 200.17\,m$$ 4. **a) Houses on the same side:** Distance between houses: $$|d_2 - d_1| = |200.17 - 114.01| = 86.16\,m$$ 5. **b) Houses on opposite sides:** Distance between houses is sum of distances: $$d_1 + d_2 = 114.01 + 200.17 = 314.18\,m$$ 6. **c) One house due South, other due East:** The houses form a right triangle with the tower at the right angle. Distance between houses by Pythagoras: $$\sqrt{d_1^2 + d_2^2} = \sqrt{114.01^2 + 200.17^2}$$ $$= \sqrt{12996.28 + 40068.03} = \sqrt{53064.31} \approx 230.38\,m$$ **Final answers:** - a) 86.16 m - b) 314.18 m - c) 230.38 m