Which of the following indicates the formula for linear method of bisection of arcs?

Question

Examoogle · Accepted Answer

Option A: $R - \sqrt{R^2 - (\frac{L}{2})^2}$

Explanation & Concepts:
In surveying, the linear method of bisection of arcs is used to locate points on a circular curve. The formula $h = R - \sqrt{R^2 - (L/2)^2}$ represents the mid-ordinate (versed sine) of a circular curve of radius $R$ and chord length $L$.

Working Principle: The principle relies on the Pythagorean theorem within a right-angled triangle formed by the radius, half the chord length, and the distance from the center to the chord. The total radius $R$ minus the perpendicular distance from the center to the chord ($sqrt{R^2 - (L/2)^2}$) yields the height of the arc segment.

Key Formulas: $h = R - \sqrt{R^2 - (\frac{L}{2})^2}$ — Formula for the mid-ordinate of a circular curve.

Key Points: The mid-ordinate is the perpendicular distance from the midpoint of the chord to the midpoint of the arc.; This method is a fundamental geometric calculation in setting out simple circular curves in highway and railway engineering.; The term $\sqrt{R^2 - (L/2)^2}$ represents the apothem, which is the distance from the center of the circle to the chord.

Advantages: Simple geometric calculation requiring only basic trigonometry or algebra.; Highly accurate for defining small segments of circular arcs in field surveys.

Disadvantages: Requires high precision when measuring long chords.; Inapplicable to non-circular curves without specific modifications.

Applications: Highway alignment and curve setting.; Railway track geometry design.; General geodetic surveying where arc-based point location is required.

Additional Info: The term under the square root, $R^2 - (L/2)^2$, must always be positive, which implies that the chord length $L$ cannot exceed the diameter $2R$.. Options C and D are incorrect because they use addition inside the square root, which does not conform to the geometry of a circular segment.

Answer: A is correct — The formula $R - \sqrt{R^2 - (L/2)^2}$ correctly calculates the mid-ordinate of a circular arc based on the Pythagorean relationship in a circle.

Exam Tip: Always verify the sign of the constant in your radical expressions for curve geometry; subtraction ($R^2 - (L/2)^2$) is standard for circular segments while addition typically relates to distance formulas.

Examoogle · Answer

Option B: $R + \sqrt{R^2 - (\frac{L}{2})^2}$

Examoogle · Answer

Option C: $R - \sqrt{R^2 + (\frac{L}{2})^2}$

Examoogle · Answer

Option D: $R + \sqrt{R^2 + (\frac{L}{2})^2}$

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