The Ambiguous Sine Rule in a Triangular Plot
A surveyor is measuring a triangular plot ABC for a client. She records AB = 7.0 m, BC = 5.0 m and angle BAC = 35°. The client wants a fence along AC and a gate at C. The surveyor's assistant uses the sine rule and reports a single value AC ≈ 8.7 m, but the surveyor suspects a second possible triangle. The plot must have an area of at least 15 m² to satisfy a planning condition, and the fence budget assumes that AC is the longest side. The surveyor must decide which configuration, if either, is consistent with all the measurements and the planning condition. Angles should be given to one decimal place, lengths to two decimal places, and area to the nearest 0.1 m². The assistant has not checked whether the angle at C is acute or obtuse, and the client is waiting for a definite answer. A scale diagram is not required, but the surveyor must justify why both mathematical solutions do or do not correspond to valid plots.
1. Using the sine rule, find the two possible sizes of angle ACB, giving angles to one decimal place. Show why two values arise.
Read the worked answer
Label the triangle in the usual way: a=BC=5 is opposite angle A=35°, and c=AB=7 is opposite angle C. The sine rule gives
sin C / c = sin A / a
sin C = c sin A / a = 7 sin 35° / 5.
Now sin 35°≈0.5736, so sin C≈7×0.5736/5≈0.8030.
Because sin C is between 0 and 1, there are two angles in the range 0° to 180° with this sine: C≈53.4° and C≈180°−53.4°=126.6°. Both must be checked for a positive third angle:
- If C=53.4°, then B=180°−35°−53.4°=91.6°.
- If C=126.6°, then B=180°−35°−126.6°=18.4°.
Both values of B are positive, so both triangles are possible at this stage. This is the ambiguous SSA case.
2. For each configuration, calculate AC and the area of the plot. Decide which configuration satisfies both the planning condition (area at least 15 m²) and the budget assumption that AC is the longest side.
Read the worked answer
Use the sine rule again for b=AC:
b / sin B = a / sin A ⇒ b=5 sin B / sin 35°.
The area is (1/2)(AB)(BC) sin B=(1/2)(7)(5) sin B=17.5 sin B.
Case 1: B=91.6°. Then AC=5 sin 91.6°/sin 35°≈5×0.9996/0.5736≈8.71 m. Area≈17.5×0.9996≈17.5 m². Here AC≈8.71 m is longer than AB=7 m and BC=5 m, so it is the longest side. The area exceeds 15 m², so this configuration meets both conditions.
Case 2: B=18.4°. Then AC=5 sin 18.4°/sin 35°≈5×0.3156/0.5736≈2.75 m. Area≈17.5×0.3156≈5.5 m². This fails the area condition, and AC=2.75 m is not the longest side because AB=7 m is longer.
Only the first configuration, with C≈53.4° and AC≈8.71 m, satisfies both the planning condition and the budget assumption.
3. Explain why the assistant's single answer AC≈8.7 m is incomplete, and state what check would reveal the second triangle.
Read the worked answer
The assistant used the sine rule and took only the acute value of C from sin C≈0.8030. Since the sine rule determines sin C, not C itself, the obtuse supplement 126.6° is also a candidate. The second triangle is genuine here because the resulting angle B=18.4° is positive, and the side lengths satisfy the triangle inequality: 2.75+5>7. The assistant's answer AC≈8.7 m corresponds only to the acute-C triangle.
A reliable check is to compute B=180°−A−C for both values of C and reject any case with B≤0. Alternatively, draw the two possible positions of side BC relative to AB. Here both positions give valid triangles, so a complete solution must present both and then use the area and longest-side conditions to select the intended plot.