Highway geometric design – perpendicular offset from tangent to the midpoint of a circular curve For a simple circular curve of radius R and central deflection angle θ, what is the perpendicular offset from the tangent to the midpoint (central point) of the curve?

Introduction / Context:
This question tests basic circular-curve geometry used in highway and railway alignment setting out. The perpendicular offset (often called the mid-ordinate from the tangent) is a common quantity in field computations and curve staking.

Given Data / Assumptions:

• Simple circular curve with radius R.
• Central angle (deflection angle) is θ.
• Offset is measured from the tangent to the midpoint of the curve (i.e., at half the deflection angle).

Concept / Approach:
At the midpoint of the curve, the line from the curve to the tangent is perpendicular. The offset equals the radial drop from the tangent to the arc and can be derived from the chord geometry for half-angle θ/2.

Step-by-Step Solution:
Consider the triangle formed by the curve center, the point of tangency, and the midpoint point on the arc.The radial line to the midpoint makes an angle θ/2 from the initial tangent direction.The perpendicular offset from the tangent equals R - R cos(θ/2).Therefore, offset = R (1 - cos θ/2).

Verification / Alternative check:
For small θ, use cos(θ/2) ≈ 1 - (θ^2/8) so the offset ≈ R * (θ^2/8), which matches small-angle arc approximations.

Why Other Options Are Wrong:

• R sin θ/2 and R cos θ/2 are raw trigonometric components, not the tangent-to-arc offset.
• R (1 - sin θ/2) mixes sine where cosine is required for the normal (perpendicular) distance.
• 'none of these' is incorrect because R (1 - cos θ/2) is standard.

Common Pitfalls:
Confusing mid-ordinate from the chord with offset from the tangent; both use θ/2 but refer to different reference lines.

Final Answer:
R (1 - cos θ/2)