Ambiguous Triangular Garden
A landscape architect is designing a triangular garden ABC. The survey fixes the corner A, the length AC = 11 m, and the angle at A = 35°. The side BC is to be 8 m long, but the surveyor has not fixed the angle at B. The client wants a garden with area at least 30 m². The architect notices that the given information may allow more than one triangle. A colleague claims that because two sides and an angle are known, the triangle is determined. The architect must check whether the angle is included, use the sine rule to find the possible angles at B, and then compute the possible areas before advising the client. All lengths are exact to the nearest metre; use degree measures.
1. Explain why the data can produce two different triangles. Support your explanation with a calculation involving the height from C to line AB.
Read the worked answer
Let a=BC=8 (opposite A=35°) and b=CA=11 (opposite B). The perpendicular distance from C to line AB is h=b sin A=11 sin35°≈11·0.5736≈6.31 m. Since the required side BC=8 m is longer than h, a circle of radius 8 centered at C meets line AB in two distinct points. Equivalently, the sine rule gives sin B=b sin A/a=11 sin35°/8≈0.7887. Both B≈52.1° and B≈127.9° are valid angles, since each leaves a positive angle C. The colleague's claim would be correct only if the given angle were included between the two given sides (SAS), which it is not.
2. Find the two possible values of angle C and the corresponding areas. Which design meets the area requirement?
Read the worked answer
For B≈52.1°, C=180°-35°-52.1°≈92.9°. Area = (1/2)ab sin C = (1/2)(8)(11) sin92.9° ≈44·0.9987≈43.9 m². For B≈127.9°, C=180°-35°-127.9°≈17.1°. Area ≈44·sin17.1°≈44·0.294≈12.9 m². Only the first triangle meets the client's requirement of at least 30 m². The second is a valid triangle but too small.
3. The client now insists that angle B be obtuse. Which triangle remains, and what is its area? Does this remove the ambiguity completely?
Read the worked answer
If angle B must be obtuse, only B≈127.9° remains. Then C≈17.1° and the area is about 12.9 m², which fails the 30 m² requirement. The obtuse-angle condition removes the ambiguity, but it selects the triangle the client does not want. The architect should therefore report that the original data allow two designs, and that the area requirement selects the acute-B design with area about 43.9 m².