Method for Double Triangle Questions
Identify the two right-angled triangles sharing a common side (or base).
Write tan θ₁ for the smaller triangle and tan θ₂ for the larger triangle.
Express the common side from equation 1 and substitute into equation 2 to solve!
Step-by-Step Problem Solving
A 1.6 m tall statue stands on top of a pedestal. From a point on ground, elevation of top of statue is 60° and top of pedestal is 45°. Find height of pedestal.
Let height of pedestal = h, distance from observer = x. Height of top of statue = h + 1.6.
In smaller △ (for pedestal):
In larger △ (for statue + pedestal):
√3 = (h + 1.6) / h (since x = h)
h√3 = h + 1.6 ⟹ h(√3 - 1) = 1.6
h = 1.6 / (√3 - 1) = 0.8(√3 + 1) metres
Height of pedestal is 0.8(√3 + 1) m (≈ 2.18 m).
As observed from top of 75 m high lighthouse, angles of depression of two ships are 30° and 45°. One ship is directly behind other. Find distance between ships.
Let height of lighthouse = 75 m. Distance to first ship = x, distance between ships = y.
In smaller △ (Ship 1):
In larger △ (Ship 2):
1 / √3 = 75 / (75 + y)
75 + y = 75√3 ⟹ y = 75(√3 - 1) metres
Distance between the two ships is 75(√3 - 1) m (≈ 54.9 m).
Key Takeaways
Always solve the 45° triangle first (since tan 45° = 1 makes Height = Distance!).
Substitute the 45° relation into the 30° or 60° triangle equation.
As an observer moves closer to a tower, the angle of elevation increases (30° ➔ 45° ➔ 60°).