Triangular Plot Survey and Storage Pad Clearance
A land surveyor measures a triangular plot ABC. Side AB is 84.0 m on a bearing of 062°, and side BC is 63.5 m on a bearing of 148°. The owner wants to fence the boundary and build a rectangular storage pad 20 m long and 12 m wide with one long side parallel to AC and set back 5 m from AC. A drainage easement requires the pad to stay at least 5 m away from vertex B. The surveyor needs to check the diagonal AC, the plot area, and whether the pad can fit without crossing the boundary. Bearings are measured clockwise from north. The ground is level. The local planning office will reject any pad that comes within 5 m of the plot boundary or of point B. The owner has not yet ordered fencing.
1. Find the internal angle at B and the length AC.
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The bearing of BA is 062° + 180° = 242°. The internal angle ABC is the difference between the bearings BA and BC: 242° − 148° = 94°. By the cosine rule: AC² = 84.0² + 63.5² − 2(84.0)(63.5)cos94°. Since cos94° ≈ −0.06976, AC² = 7056 + 4032.25 + 744.2 = 11832.45, so AC ≈ 108.8 m.
2. Calculate the area of the plot and the shortest distance from B to AC.
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Area = 0.5 × 84.0 × 63.5 × sin94° ≈ 0.5 × 5334 × 0.99756 ≈ 2660.5 m². The shortest distance from B to AC is the height h: area = 0.5 × AC × h, so h = 2 × 2660.5 / 108.78 ≈ 48.9 m. The foot of the perpendicular is at distance from A: (AB² + AC² − BC²) / (2AC) = (7056 + 11832.45 − 4032.25) / 217.56 ≈ 68.3 m.
3. Determine whether the 20 m × 12 m pad can fit with its long side parallel to AC, a 5 m setback from AC, and at least 5 m clearance from B. Show the limiting checks.
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The pad needs 5 m setback plus 12 m width = 17 m perpendicular clearance from AC. Check a 20 m length centred on the foot of the perpendicular: at 10 m either side, the available heights are h(58.3) = 48.9 × (58.3 / 68.3) ≈ 41.7 m and h(78.3) = 48.9 × ((108.8 − 78.3) / (108.8 − 68.3)) ≈ 36.8 m. Both exceed 17 m, so the pad fits within the triangle and leaves at least 36.8 − 17 ≈ 19.8 m to B, above the 5 m easement. The limiting check is the end of the 20 m segment nearer C, not the perpendicular foot. So the pad can fit, provided the 20 m length is kept within those bounds.