The Surveyor's Ambiguous Triangle
A surveyor measures a triangular plot ABC from point B. The baseline BA is 7.00 m, the distance AC is 5.00 m, and angle ABC is 30°. The assistant computes sin C = (7 sin 30°)/5 = 0.7, takes the inverse sine, and reports a single triangle with C about 44.4°. The client needs BC and the area to order fencing and grass seed. A second surveyor warns that the measurements may allow two different plots, so the report could understate the fencing and seed needed. The site plan requires every configuration consistent with the measurements, with side lengths and areas rounded to two decimal places.
1. How many triangles satisfy the measurements? Justify with the sine rule and the angle-sum condition.
Read the worked answer
By the sine rule, sin C/AB = sin B/AC, so sin C = AB sin B / AC = 7·0.5/5 = 0.7. On (0°,180°), sin C=0.7 gives two candidates:
C1≈44.43° and C2=180°−44.43°=135.57°.
With B=30°, the remaining angles are A1≈105.57° and A2≈14.43°, both positive and both summing to 180° with B and C. Therefore there are two noncongruent triangles, not one.
2. For each configuration, compute BC and the area. Which one should be used if the client requires area at least 10 m²?
Read the worked answer
The sine rule gives BC/sin A = AC/sin B = 5/0.5 = 10, so BC=10 sin A.
Case 1: A≈105.57°, BC≈10(0.9634)=9.63 m. Area = (1/2)(BA)(BC) sin B = 0.5·7·BC·0.5 = 1.75 BC ≈16.86 m².
Case 2: A≈14.43°, BC≈10(0.2492)=2.49 m. Area ≈1.75(2.49)=4.36 m².
The area-at-least-10 m² requirement admits only the first configuration.
3. The assistant claims that because sin C=0.7 has a unique value, the triangle is unique. Identify the logical error and state a general condition under which the ambiguous case gives two triangles.
Read the worked answer
The error is treating sine as one-to-one on (0°,180°). A single sine value can correspond to two supplementary angles.
In the SSA setting with given angle B, opposite side b=AC=5, and adjacent side c=AB=7, the height is h=c sin B=3.5. Since h<b<c, the side b can swing to two positions, giving two triangles. In general, for SSA, two triangles occur when c sin B < b < c; here 3.5<5<7, confirming the ambiguous case.