Problem Statement
A chain line PQR crosses a stream, Q and R being the near and far off banks respectively. A line QM of length 60 m is set out at right angles to the chain line at Q. If the bearings of QM and MR are 282° 45′ and 42° 45′ respectively, find the width of the stream.
Step-by-Step Solution
Key Information
- Chain line PQR crosses stream (Q near bank, R far bank).
- Line QM = 60 metres.
- QM is perpendicular to PQR at Q (∠RQM = 90°).
- Bearing of QM = 282° 45′.
- Bearing of MR = 42° 45′.
- Goal: Find the width of the stream (distance QR).
Step 1: Calculate Back Bearing of MQ
The bearing of MQ (from M back to Q) is needed to calculate angles at M. It differs from the bearing of QM by 180°.
Bearing of MQ = Bearing of QM ± 180°
Since Bearing QM > 180°, we subtract 180°:
Bearing of MQ = 282° 45′ − 180°
Bearing of MQ = 102° 45′
Step 2: Calculate Angle RMQ
The angle RMQ at point M is the difference between the bearing of MQ and the bearing of MR.
∠RMQ = Bearing of MQ − Bearing of MR
∠RMQ = 102° 45′ − 42° 45′
∠RMQ = 60°
Step 3: Calculate Width of Stream (QR)
We are given that QM is set out at right angles to the chain line PQR at Q. Therefore, triangle RQM is a right-angled triangle with the right angle at Q (∠RQM = 90°).
In right-angled triangle RQM, we know:
– Side QM (Adjacent to ∠RMQ) = 60 m
– Angle RMQ = 60°
– Side QR (Opposite to ∠RMQ) is the stream width we need to find.
Using the tangent relationship:
tan(∠RMQ) = Opposite / Adjacent = QR / QM
Rearranging to find QR:
QR = QM × tan(∠RMQ)
QR = 60 m × tan(60°)
QR = 60 m × 1.73205…
QR ≈ 103.92 metres
Final Result
Conceptual Explanation & Applications
Core Concepts:
- Chain Line: The main survey line being measured or followed.
- Bearings: Clockwise angles from North used to define the direction of lines (QM, MR).
- Back Bearings: The bearing in the opposite direction along a line (MQ vs QM), differing by 180°. Crucial for calculating angles at the far end of a line.
- Perpendicular Offset: Setting out a line (QM) at exactly 90° to the main survey line (PQR) at a known point (Q). This creates a right-angled triangle.
- Right Triangle Trigonometry: Using trigonometric functions (sine, cosine, tangent) to relate angles and side lengths in right-angled triangles (like ΔRQM). In this case, tangent was used as we knew the adjacent side (QM) and an angle (∠RMQ), and needed the opposite side (QR).
Real-World Applications:
- Measuring Inaccessible Distances: Standard technique for finding widths of rivers, streams, canyons, or other obstacles without needing to physically cross them with measuring equipment.
- Site Layout: Determining building setbacks or distances to features from a baseline.
- Infrastructure Planning: Estimating material needs (e.g., bridge length, pipe length) for crossing obstacles.
- Boundary Surveys: Locating boundary corners or lines separated by inaccessible terrain or water bodies.
Why It Works:
This method relies on creating a solvable geometric shape – a right-angled triangle. By setting out a line QM of known length (60m) perpendicular to the stream bank at Q, we establish one leg and the right angle of triangle RQM.
Measuring the bearings of QM and MR allows us to calculate the internal angle RMQ at the vertex M using the concept of back bearings. With one leg (QM) and one acute angle (∠RMQ = 60°) known in the right-angled triangle RQM, we can use the tangent function (tan = opposite/adjacent) to calculate the length of the other leg (QR), which represents the width of the stream.
This approach provides an accurate way to measure the stream’s width using measurements taken entirely from the near bank (point Q) and the offset point (M).
