The Ambiguous Lot: SSA, Frontage, and Valuation
A surveyor maps a triangular lot and measures side AB = 120 ft, side BC = 90 ft, and the angle at A (between AB and AC) as 28°. The lot's road frontage lies along side AC, and the county requires at least 30 ft of frontage, while the neighboring parcel's fence line leaves at most 60 ft usable. The surveyor's assistant uses the Law of Sines to compute angle C, reports a single value of 38.8°, then computes AC ≈ 176 ft and concludes the lot cannot be developed. The surveyor suspects the assistant has mishandled the given information and asks for a full analysis before the lot is appraised at $30 per square foot.
1. Determine all values of angle C consistent with the measurements and explain why more than one value arises from the same data.
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Law of Sines. sin C / AB = sin A / BC, so sin C = AB · sin A / BC = 120 · sin 28° / 90 = 120(0.4695)/90 = 0.6260.
Two solutions. C = arcsin(0.6260) = 38.8°, or C = 180° − 38.8° = 141.2°. Both are geometrically possible because the given side-angle-side-adjacent pattern is the ambiguous SSA case: sin C has the same value for supplementary angles.
Check the third angle. With C = 38.8°, B = 180° − 28° − 38.8° = 113.2° > 0. With C = 141.2°, B = 180° − 28° − 141.2° = 10.8° > 0. Both triangles are valid, so the measurements genuinely permit two lots. The assistant reported only the acute solution.
2. Compute AC for both configurations and identify which one satisfies the frontage constraints.
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Law of Sines for AC. AC / sin B = BC / sin A → AC = 90 sin B / sin 28°.
Triangle 1 (B = 113.2°). sin 113.2° = 0.9191, so AC = 90(0.9191)/0.4695 ≈ 176.2 ft.
Triangle 2 (B = 10.8°). sin 10.8° = 0.1874, so AC = 90(0.1874)/0.4695 ≈ 35.9 ft.
Constraint check. Required frontage is 30–60 ft along AC. Only 35.9 ft lies in that interval; 176.2 ft exceeds the 60-ft usable limit.
Conclusion. The lot is developable, but only in the obtuse-C configuration (C = 141.2°, B = 10.8°). The assistant selected the wrong member of the ambiguous pair, so the conclusion is not supported.
3. For the developable configuration, compute the lot's area and assess the assistant's implicit assumption that valuation is unaffected by which triangle is used.
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Area formula. Area = ½ · AB · BC · sin B, using the included angle at B.
Developable triangle (B = 10.8°). Area = ½(120)(90)(0.1874) = 5400(0.1874) ≈ 1,012 ft². At $30 per ft² the value is about $30,360.
The other triangle (B = 113.2°). Area = 5400(0.9191) ≈ 4,963 ft², or about $148,900.
Assessment. The two configurations differ in area by a factor of about 4.9, so the choice of triangle changes the appraisal by roughly $118,500. The assistant's single-solution shortcut would both misstate the value and contradict the frontage limit. Any valuation must first resolve the ambiguity using physical evidence, such as a measured diagonal or the recorded fence line.