Surveying Across a Ravine: The Ambiguous SSA Case
A surveyor needs the distance BC across a ravine. Standing at B, the surveyor measures BA = 25 m and angle ABC = 35 degrees. An assistant at C measures CA = 18 m. The surveyor sets up the Law of Sines using side b = CA = 18 m opposite angle B, and side c = BA = 25 m opposite angle C. A field note from a previous survey says the ravine is less than 15 m wide at this crossing, but the surveyor is not sure the note applies. The surveyor computes sin C ≈ 0.7967 and is about to report a single distance BC. A colleague warns that the measurements may allow more than one triangle, so the report must state which configuration is being used and why.
1. Find both possible values of angle C and the corresponding values of angle A. Explain why two triangles satisfy the given measurements.
Read the worked answer
By the Law of Sines, sin C / c = sin B / b, so
sin C = c sin B / b = 25 sin 35 degrees / 18 ≈ 25(0.5736)/18 ≈ 0.7966.
The inverse sine gives C ≈ 52.8 degrees. But sine is also positive in quadrant II, so C = 180 - 52.8 = 127.2 degrees is also possible. Then A = 180 - 35 - C:
If C ≈ 52.8 degrees, A ≈ 92.2 degrees. If C ≈ 127.2 degrees, A ≈ 17.8 degrees.
This is the ambiguous SSA case: two sides and a non-included angle are given, and the side opposite the given angle is shorter than the other given side but long enough to close the triangle in two different ways. Both triangles satisfy the measurements.
2. Compute BC for both configurations. Then use the ravine note to select the reportable distance and justify the choice.
Read the worked answer
BC is side a, opposite angle A. Using the Law of Sines,
a = b sin A / sin B = 18 sin A / sin 35 degrees.
For A ≈ 92.2 degrees: a ≈ 18(0.9993)/0.5736 ≈ 31.4 m. For A ≈ 17.8 degrees: a ≈ 18(0.3057)/0.5736 ≈ 9.60 m.
The field note says the crossing is less than 15 m wide, so the reportable distance is BC ≈ 9.60 m. The 31.4 m solution corresponds to the obtuse-angle configuration, which the ravine note rules out. The report should state both possibilities and then select the 9.60 m triangle using the field note.
3. Compute the area of the selected triangle and verify it with an independent method.
Read the worked answer
For the selected triangle, use Area = (1/2)bc sin A with b = 18, c = 25, and A ≈ 17.8 degrees:
Area = 0.5(18)(25) sin 17.8 degrees ≈ 225(0.3057) ≈ 68.8 m^2.
Check with Heron's formula. The sides are 18, 25, and 9.60, so s = (18 + 25 + 9.60)/2 = 26.3. Then
Area = sqrt(26.3(26.3 - 18)(26.3 - 25)(26.3 - 9.60)) = sqrt(26.3 * 8.3 * 1.3 * 16.7) ≈ sqrt(4738) ≈ 68.8 m^2.
The two methods agree, supporting the 9.60 m choice.