[{"type":"technical","methodBadge":"Concept & Principle","explanation":"In simple circular curve survey, the long chord is the straight line segment joining the point of curve (BC) to the point of tangency (EC). Its length is determined by the radius of the curve and the deflection (or central) angle. Using the standard curve relation $L = 2R \\sin(\Delta / 2)$, for a radius of $25\text{ m}$ and deflection angle of $100°$, the calculated long chord length is $38.302\text{ m}$.","workingPrinciple":"The center of the curve forms an isosceles triangle with the two tangent points, subtending an angle equal to the total deflection angle $\Delta$. A perpendicular dropped from the center to the long chord bisects both the chord and the central angle into two right-angled triangles. Applying trigonometric sine ratio gives half the chord length as $R \\sin(\Delta / 2)$, making the total length $2R \\sin(\Delta / 2)$.","diagramSvg":"Long Chord FormulaL = 2 × R × sin(Δ / 2)

• L = 2 × 25 × sin(50°) = 50 × 0.76604 = 38.302 m","keyFormulas":["$L = 2R \\sin\left(\frac{\Delta}{2}\right)$ — Length of Long Chord","$T = R \\tan\left(\frac{\Delta}{2}\right)$ — Tangent Length of Simple Curve","l = \frac{\pi R \Delta}{180° — Length of Curve Arc"],"keyPoints":["The deflection angle $\Delta$ at the intersection point is equal to the central angle subtended by the curve arc at the center.","The long chord represents the shortest distance between the Point of Curve (PC) and Point of Tangency (PT).","When $\Delta = 180°$, the long chord equals the diameter of the circle ($2R$)."],"advantages":[],"disadvantages":[],"applications":["Setting out simple circular curves in road and railway alignments.","Verification of curve geometry during field surveying using total station or transit instruments."],"comparisonTable":[],"additionalInfo":["Option A ($49.240\text{ m}$) corresponds to the curve arc length: $l = \frac{\pi \times 25 \times 100}{180} \approx 43.63\text{ m}$ or tangent sum error.","Option C ($59.587\text{ m}$) corresponds to using incorrect trigonometric relations such as $2R \\tan(\Delta/2) = 2(25)\\tan(50°) = 59.588\text{ m}$, which gives the sum of tangent lengths ($2T$).","Option D ($32.139\text{ m}$) represents $2R \\cos(\Delta/2) = 50 \\cos(50°) = 32.139\text{ m}$."],"answer":"B is correct — Applying $L = 2R \\sin(\Delta / 2)$ with $R = 25\text{ m}$ and $\Delta = 100°$ gives $L = 2(25)\\sin(50°) = 38.302\text{ m}$.","correctedOptions":["A) 49.240 m","B) 38.302 m","C) 59.587 m","D) 32.139 m"],"coreConcepts":["Simple Circular Curve Geometry","Long Chord Calculation","Deflection Angle & Radius Relationship"],"crossTopicTip":"Always distinguish between Long Chord $L = 2R\\sin(\Delta/2)$, Tangent Length $T = R\\tan(\Delta/2)$, and Arc Length $l = R\\Delta_{\text{rad}}$ during surveying exam calculations."},"For a visual step-by-step tutorial on solving horizontal curve parameters for engineering exams","watch",["How to Find Chord Length on a Horizontal Curve"],"(https:"]