Problem Statement
A survey line CDE crosses a river, D being on the near bank, and E on the opposite bank. A perpendicular DF = 150 metres is ranged at D on the left. From F bearings of E and C are observed to be 25° and 115° respectively. If the chainage of C is 1250 metres and that of D is 1620 metres, find the chainage of E.
Step-by-Step Solution
Key Information
- Survey line CDE crosses river (D near bank, E far bank).
- Perpendicular distance DF = 150 metres (ranged at D).
- Bearing of FE = 25°.
- Bearing of FC = 115°.
- Chainage of C = 1250 metres.
- Chainage of D = 1620 metres.
- Goal: Find the chainage of E.
Step 1: Calculate Length CD
Chainage represents the distance along the survey line from a starting point. The length of segment CD is the difference between the chainages of D and C.
Length CD = Chainage of D − Chainage of C
Length CD = 1620 m − 1250 m
Length CD = 370 m
Step 2: Calculate Angle EFC
The angle EFC at point F can be found by the difference between the bearings of FC and FE, measured from F.
∠EFC = Bearing of FC − Bearing of FE
∠EFC = 115° − 25°
∠EFC = 90°
This means triangle EFC is a right-angled triangle, with the right angle at F.
Step 3: Calculate Distance DE (River Width)
We are given that DF is perpendicular to the survey line CDE at D. This means ∠FDC = 90° and ∠FDE = 90°.
Since ∠FDC = 90°, triangle FDC is right-angled at D.
Since ∠FDE = 90°, triangle FDE is right-angled at D.
From Step 2, we know triangle EFC is right-angled at F.
In a right-angled triangle EFC, if DF is the altitude to the hypotenuse CE (which holds true given DF⊥CE and ∠EFC=90°), a geometric property states: FD² = ED × DC.
Rearranging to find ED:
ED = FD² / DC
ED = (150 m)² / 370 m
ED = 22500 m² / 370 m
ED ≈ 60.81 metres
(This value represents the width of the river segment DE along the survey line).
Step 4: Calculate Chainage of E
The chainage of E is found by adding the distance DE to the chainage of D.
Chainage of E = Chainage of D + Length DE
Chainage of E = 1620 m + 60.81 m
Chainage of E = 1680.81 m
Final Result
Conceptual Explanation & Applications
Core Concepts:
- Chainage: A measurement of distance along a survey line, route, or alignment, typically starting from zero. Used extensively in road, rail, and pipeline projects.
- Bearings & Angles: Using bearings (angles from North) to determine the angles between survey lines (like ∠EFC).
- Perpendicular Offsets: Establishing a line (DF) perpendicular to the main survey line (CDE) to create right-angled triangles for calculation.
- Right Triangle Geometry: Applying properties of right triangles, including trigonometric relationships and geometric theorems (like the altitude theorem where FD² = ED × DC in this specific configuration).
Real-World Applications:
- Route Surveying: Determining exact locations (chainages) of points along linear projects like roads, railways, canals, or pipelines, especially when crossing obstacles like rivers.
- Bridge & Culvert Placement: Accurately locating start and end points for structures crossing obstacles.
- Construction Layout: Setting out points based on calculated chainages for construction activities.
- Topographic Mapping: Mapping features relative to established survey lines and chainages.
Why It Works:
This problem combines the concepts of chainage and indirect measurement using bearings and perpendiculars. By knowing the chainages of C and D, we establish the length of the segment CD. Setting up a perpendicular DF allows us to use point F as a reference. Measuring bearings from F to C and E enables the calculation of angle EFC.
The crucial step involves recognizing the geometric relationships formed. The perpendicular DF creates right triangles FDC and FDE. The calculated angle EFC being 90° confirms triangle EFC is also a right triangle. This specific geometry allows the use of the relationship FD² = ED × DC to find the unknown river width DE along the survey line.
Finally, adding the calculated distance DE to the known chainage of D gives the required chainage of point E on the far bank. This method is practical for determining precise locations along a route when direct measurement across obstacles is not feasible.