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Difficulty: HardSine and Cosine Rules

A surveyor at station AA observes two landmarks, BB and CC. Landmark BB is located at a distance of 14 km14\text{ km} from AA on a bearing of 025∘025^\circ. Landmark CC is located at a distance of 62 km6\sqrt{2}\text{ km} from AA on a bearing of 070∘070^\circ. What is the direct distance between landmark BB and landmark CC?

  1. 10 km10\text{ km}Answer
  2. B
    72 km7\sqrt{2}\text{ km}
  3. C
    8 km8\text{ km}
  4. D
    246 km2\sqrt{46}\text{ km}

Answer

The direct distance between landmark BB and landmark CC is 10 km10\text{ km}.
The included angle ∠BAC\angle BAC between the two bearings is 070∘−025∘=45∘070^\circ - 025^\circ = 45^\circ. Applying the Cosine Rule a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A gives a2=(62)2+142−2(62)(14)cos⁡45∘=72+196−168=100a^2 = (6\sqrt{2})^2 + 14^2 - 2(6\sqrt{2})(14)\cos 45^\circ = 72 + 196 - 168 = 100. Taking the square root gives 10 km10\text{ km}.

Step-by-Step Solution

1
Determine the interior angle ∠BAC\angle BAC from the given bearings.
∠BAC=070∘−025∘=45∘\angle BAC = 070^\circ - 025^\circ = 45^\circ
The difference between two bearings measured clockwise from North from the same point gives the included angle between the lines of sight.
2
Identify the side lengths adjacent to angle AA in ΔABC\Delta ABC.
c=AB=14 kmc = AB = 14\text{ km}, b=AC=62 kmb = AC = 6\sqrt{2}\text{ km}, and included angle A=45∘A = 45^\circ
We have a Side-Angle-Side (SAS) triangle configuration, requiring the Cosine Rule to find the opposite side a=BCa = BC.
3
Apply the Cosine Rule a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A.
a2=(62)2+142−2(62)(14)cos⁡45∘a^2 = (6\sqrt{2})^2 + 14^2 - 2(6\sqrt{2})(14)\cos 45^\circ
Substituting known values into the Cosine Rule formula.
4
Simplify the terms and solve for aa.
a2=72+196−1682(12)=268−168=100  ⟹  a=100=10 kma^2 = 72 + 196 - 168\sqrt{2}\left(\frac{1}{\sqrt{2}}\right) = 268 - 168 = 100 \implies a = \sqrt{100} = 10\text{ km}
Squaring 626\sqrt{2} gives 36×2=7236 \times 2 = 72, 142=19614^2 = 196, and simplifying the cosine term yields 168168.

Key Concept

Cosine Rule for SAS non-right triangles in bearing contexts
Estimated Time:2m 0s
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Topics

Sine and Cosine Rules

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