From each vertex A, B, C of a horizontal triangle ABC, the angle of elevation of a hilltop is α (same at all three vertices). If side a is opposite A, find the height of the hill in terms of a and α.

Introduction / Context:If the angle of elevation to the same point is equal from all three vertices of triangle ABC, the foot of the perpendicular from the hilltop to the plane lies at the circumcenter O of triangle ABC. The horizontal distance from O to each vertex equals the circumradius R, making the 3D geometry manageable.

Given Data / Assumptions:

• Elevation = α from A, B, C.
• Let hill height be h; horizontal distance from each vertex to foot O is R (circumradius).

Concept / Approach:At any vertex (say A), tan α = h / AO = h / R ⇒ h = R tan α. Triangle relation: a = 2R sin A ⇒ R = a / (2 sin A). Substitute for R.

Step-by-Step Solution:

Verification / Alternative check:Using b and B or c and C yields analogous forms (b tan α csc B / 2, etc.). All are consistent because a = 2R sin A, etc.

Why Other Options Are Wrong:(a) and (c) omit the division by 2 that comes from R = a/(2 sin A); options mixing sec with A are not from the circumradius identity.

Common Pitfalls:Assuming centroid or incenter instead of circumcenter; forgetting that equal elevations imply equal horizontal distances.

Final Answer:a tan α cosec A / 2