Two towers of height h₁ and h₂ subtend angles of 30° and 60° respectively at a point on the line joining their feet. If the distance between the towers is d, what is the ratio of their heights h₁ : h₂ if the point is at the midpoint?

Solution

Solution: Since the point is midway between the towers, the horizontal distance to each tower is d/2. Therefore, \(h_1=\frac d2\tan30^\circ=\frac{d}{2\sqrt3}\), and \(h_2=\frac d2\tan60^\circ=\frac{\sqrt3d}{2}\). Hence, \(h_1:h_2=\frac1{\sqrt3}:\sqrt3=1:3\). Trick: At the midpoint, the horizontal distances are equal, so the height ratio is the ratio of the tangents of the angles.