AI Tutor
How do surveyors measure the height of a mountain peak when a rushing river prevents them from ever reaching its base?
The solution is to take two ground measurements from different distances, decomposing the scene into a pair of overlapping right triangles.
How do we link two triangles when we cannot measure the full base of either one?

The Shared-Height Foundation

In any right triangle, the tangent ratio relates an acute angle θ\theta to the opposite vertical side and adjacent horizontal side: tan⁑(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}.
Over two thousand years ago, Greek astronomers like Aristarchus used shadow proportions to measure distances across the solar system, discovering that identical angle ratios govern right triangles of every size.
What happens when the distance to the base is an unknown variable xx?

Inaccessible Base: The Two-Station Setup

Suppose you stand at Station A and measure an angle of elevation of 30∘30^\circ to a cliff's summit. You then walk 100 ft100\text{ ft} closer to Station B and measure an angle of 45∘45^\circ.
Let hh represent the unknown cliff height, and let xx be the unknown horizontal distance from Station B to the base of the cliff.
From the inner right triangle at Station B, we write tan⁑(45∘)=hx\tan(45^\circ) = \frac{h}{x}. Since tan⁑(45∘)=1\tan(45^\circ) = 1, this gives us the relation x=hx = h.