Circular Curves MCQ with answer and solution

Concept

Angle of intersection I = 180° − deflection angle Δ. Long chord L = 2R sin(Δ/2) and tangent length T = R tan(Δ/2).

Formula

Δ = 180° − I; T = R tan(Δ/2); L = 2R sin(Δ/2); L/T = 2cos(Δ/2)

Step-by-step solution

1. Angle of intersection I = 120°, so deflection angle Δ = 180° − 120° = 60°. 2. Tangent length T = R tan(30°) = R/√3. 3. Long chord L = 2R sin(30°) = R. 4. L/T = R ÷ (R/√3) = √3.

Explanation

The angle of intersection is the angle between the two straights, so the deflection angle is 60°. Then L/T = 2cos(Δ/2) = 2cos30° = √3.

Why the other options are wrong

A: 1 comes from wrongly taking 120° as the deflection angle (2cos60° = 1). B: 1/√3 is the tangent ratio tan30°, not the chord-to-tangent ratio. C: 0.866 is cos30° alone, which misses the factor 2.

Shortcut

L/T = 2cos(Δ/2), with Δ = 180° − I.

Common mistake

Using the angle of intersection directly as the deflection angle.