Circular-curve geometry: when long chord equals tangent length In a simple circular curve of radius R connecting two tangents, suppose the long chord (between the tangent points) is equal to the tangent length. For this case, what is the intersection/deflection angle Δ of the curve?

Introduction / Context:
In highway and railway geometric design, several geometric quantities for a simple circular curve are related to the intersection (deflection) angle Δ and radius R. Two key quantities are the tangent length T and the long chord LC. This question tests your ability to use the standard relationships to determine Δ when the long chord equals the tangent length.

Given Data / Assumptions:

• Simple circular curve of radius R connects two tangents.
• Tangent length T = R * tan(Δ/2).
• Long chord LC = 2 * R * sin(Δ/2).
• Given condition: LC = T.

Concept / Approach:

Use the standard formulas for a simple circular curve. Equating the long chord to the tangent length yields an equation in Δ/2 that can be solved using basic trigonometric identities, specifically the relationship between sine, cosine, and tangent for the same angle.

Step-by-Step Solution:

Verification / Alternative check:

Plug Δ = 120° into formulas: T = R * tan 60° = R * √3; LC = 2R * sin 60° = 2R * (√3/2) = R * √3. Hence LC = T, confirming the result.

Why Other Options Are Wrong:

30°, 60°, 90°, 150° do not satisfy LC = T when substituted; each produces LC and T values that differ.

Common Pitfalls:

Confusing long chord with normal chord; mixing up T = R * tan(Δ/2) and external distance; forgetting to reject the trivial solution Δ = 0.

Final Answer:

120°