The Ambiguous Triangle Survey
A surveyor measures a triangular plot ABC. Side AB = 8.0 m, side AC = 5.0 m, and angle ABC (at vertex B) = 30°. She wants the length of BC and the area of the plot. A colleague warns that the given information may not determine a unique triangle. The surveyor first applies the sine rule: sin C / 8 = sin 30° / 5, giving sin C = 0.8. She computes C ≈ 53.13° and then finds angle A = 180° − 30° − 53.13° = 96.87°, before using the cosine rule for BC.
1. Explain why the surveyor's data produce two possible triangles and find both possible values of angle C and angle A.
Read the worked answer
The sine rule gives sin C = 8 sin 30° / 5 = 0.8. The equation sin C = 0.8 has two solutions in 0°–180°: C ≈ 53.13° and C ≈ 180° − 53.13° = 126.87°. Both are geometrically valid because the side opposite the given 30° angle (AC = 5) is shorter than AB = 8, the classic ambiguous case (SSA).
- Case 1: C ≈ 53.13°, so A = 180° − 30° − 53.13° = 96.87°.
- Case 2: C ≈ 126.87°, so A = 180° − 30° − 126.87° = 23.13°.
Both triangles satisfy the given measurements, so the data are ambiguous.
2. Compute BC and the area for both cases, and advise which case a surveyor should prefer if the plot is known to be acute-angled.
Read the worked answer
Use the cosine rule with BC² = AB² + AC² − 2·AB·AC·cos A.
Case 1 (A ≈ 96.87°): cos A ≈ −0.1197. BC² = 64 + 25 − 2·8·5·(−0.1197) = 89 + 9.58 ≈ 98.58, so BC ≈ 9.93 m. Area = ½·AB·AC·sin A = ½·8·5·sin 96.87° ≈ 20·0.9928 ≈ 19.86 m².
Case 2 (A ≈ 23.13°): cos A ≈ 0.9196. BC² = 89 − 2·8·5·0.9196 = 89 − 73.57 ≈ 15.43, so BC ≈ 3.93 m. Area = 20·sin 23.13° ≈ 20·0.3928 ≈ 7.86 m².
If the plot is known to be acute-angled, angle C ≈ 53.13° (Case 1) is acute, but angle A ≈ 96.87° is obtuse - so Case 1 is not acute. Case 2 has C obtuse. Neither is fully acute, so the acute-angled condition cannot be met; the surveyor must obtain an extra measurement (e.g. BC or angle A) to resolve the ambiguity.