Calculate the length (m) of the longer chord of a 250 m radius curve having deflection angle of 90 degrees.

Detailed Explanation: Length of the Long Chord

The long chord is the straight-line distance directly connecting the point of curvature (PC) and the point of tangency (PT) — distinct from the curve length (Q10), which follows the curved arc instead. This is a direct, single-formula application.

Solution:
The long chord formula is:

L = 2R sin(Δ/2)

Given R = 250 m, Δ = 90°:
L = 2 × 250 × sin(45°) = 500 × 0.7071 = 353.55 m ≈ 353.6 m

Why the Other Options Are Wrong

A — 250: This is simply the radius value itself, not the long chord — a clear sign of an incomplete or incorrect formula application.

C — 392.7: Does not correspond to the correct chord formula result; may come from confusing the long chord formula with the curve length or apex distance formula instead.

D — 500: This is exactly 2R (the diameter) — the result of forgetting to multiply by sin(45°) and simply using the maximum possible chord value (which would only be correct for a full 180° deflection, i.e., a semicircle).

Key Concepts for Students

• Long chord formula uses sine, not tangent or cosine: L = 2Rsin(Δ/2), distinct from the tangent length formula T = Rtan(Δ/2) (Q21, Q25) and the versed sine formula M = R[1−cos(Δ/2)] (Q1) — three different curve elements, three different trigonometric functions.
• At Δ=180°, the long chord equals the diameter: sin(90°)=1, so L=2R exactly — this is the special case (a full semicircular curve) where the long chord reaches its maximum possible value, explaining why option D (500 = 2×250) is a tempting but incorrect distractor for this 90° (not 180°) problem.
• This same question's ratio appears conceptually in Q5: Where the ratio of long chord to tangent length was calculated using the same 2Rsin(Δ/2) numerator — recognising the shared formula across multiple questions reinforces the underlying geometric relationships.