Surveying a Field Corner Without a Right Angle
A land surveyor is measuring a triangular field corner. From a fixed peg P, she measures two boundary lines: PQ = 85.0 m and PR = 64.0 m, with the included angle at P recorded as 118°. She needs the length of the third boundary QR and the area of the triangle for a land-registration form. Her assistant, working from a rough sketch, insists the triangle is 'nearly a right triangle' and proposes using QR² ≈ 85² + 64² and area ≈ ½ × 85 × 64. The surveyor suspects both shortcuts are wrong because the angle is obtuse, not 90°. Instrument tolerance is ±0.5° on the angle and ±0.2 m on each length; the registration form requires the area to the nearest square metre and the length to the nearest 0.1 m.
1. Compute QR correctly and show numerically why the assistant's Pythagorean shortcut fails.
Read the worked answer
Use the cosine rule: QR² = PQ² + PR² − 2·PQ·PR·cos(118°).
- 85² = 7225; 64² = 4096; sum = 11321.
- cos 118° ≈ −0.46947.
- 2 × 85 × 64 = 10880; 10880 × (−0.46947) ≈ −5107.8.
- So QR² ≈ 11321 − (−5107.8) = 16428.8.
- QR ≈ √16428.8 ≈ 128.2 m.
The assistant's value would be √11321 ≈ 106.4 m - about 22 m too short. The error arises because cos 118° is negative, so the −2ab·cos C term is positive and adds to the sum of squares; ignoring it is only valid at 90°. The obtuse angle opens the triangle out, making QR longer than either given side.
2. Compute the area of the triangle and state it to the nearest square metre, with a brief comment on how sensitive it is to the ±0.5° angle tolerance.
Read the worked answer
Use Area = ½·PQ·PR·sin(118°).
- ½ × 85 × 64 = 2720.
- sin 118° ≈ 0.88295.
- Area ≈ 2720 × 0.88295 ≈ 2401.6 m², i.e. 2402 m² to the nearest square metre.
The assistant's ½ × 85 × 64 = 2720 m² overstates the area by about 318 m², again because it assumes a right angle where sin 90° = 1.
Sensitivity: d(Area)/dθ = 2720·cos θ. At θ = 118°, |cos θ| ≈ 0.469, so a ±0.5° (±0.00873 rad) error changes the area by about 2720 × 0.469 × 0.00873 ≈ ±11 m². That is larger than the rounding unit, so the surveyor should record the area as approximately 2402 m² with a stated uncertainty of roughly ±11 m² rather than implying exactness.