Ambiguous Triangular Plot for a Nursery
A triangular plot ABC is being surveyed for a small nursery. The surveyor measures AB = 80 m, BC = 50 m, and angle BAC = 30°. The contractor computes one area and submits a layout. However, the survey notes do not state whether angle ACB is acute or obtuse. A drainage channel must run along the third side AC, and the local authority requires this channel to be at least 70 m long. The contractor's plan shows AC as about 39 m, but the nursery owner believes a second, larger triangular plot is possible with the same measurements. The surveyor must determine how many triangles satisfy the given data, calculate the corresponding areas, and decide which configuration meets the drainage requirement without altering the measured sides or angle.
1. Use the sine rule to determine how many distinct triangles satisfy AB=80 m, BC=50 m and angle A=30°. Find angle C and angle B for each.
Read the worked answer
The sine rule gives BC/sin A = AB/sin C, so 50/sin30° = 80/sin C. Thus sin C = 80×0.5/50 = 0.8. Hence C=53.13° or C=180°−53.13°=126.87°. Both are compatible with A=30° because each leaves a positive angle B. If C=53.13°, then B=180−30−53.13=96.87°. If C=126.87°, then B=180−30−126.87=23.13°. Therefore two distinct triangles satisfy the data.
2. For each triangle, calculate AC and the area. Then decide which triangle the nursery can use if the drainage channel along AC must be at least 70 m long.
Read the worked answer
Use AC/sin B = BC/sin A = 50/0.5 = 100, so AC=100 sin B. Case 1: B=96.87°, sin B≈0.9928, so AC≈99.3 m and area=(1/2)(80)(50) sin B≈2000×0.9928≈1985.6 m². Case 2: B=23.13°, sin B≈0.3928, so AC≈39.3 m and area≈785.6 m². Only the first case has AC≥70 m. The nursery must use the triangle with C obtuse, C≈126.87°, giving AC≈99.3 m.
3. Explain why the contractor's plan showing AC≈39 m cannot be approved even though it uses the measured values.
Read the worked answer
The measurements AB=80 m, BC=50 m and angle A=30° form the ambiguous SSA case. The sine rule gives two valid values of C, hence two non-congruent triangles. The plan showing AC≈39 m corresponds to the acute-C triangle, but it fails the drainage condition AC≥70 m. Since the survey notes do not rule out the obtuse-C triangle, the contractor must not treat the 39 m layout as the unique solution.