A triangular field ABC has side AB = 85 m and side AC = 110 m. The angle BAC is 55°.
(a) Calculate the length of side BC. [3]
(b) Calculate the area of the field ABC. [3]
A vertical flagpole, BT, is erected at point B. The angle of elevation from A to the top of the flagpole T is 15°.
(c) Calculate the height of the flagpole, BT. [2]
(d) Calculate the angle of elevation from C to the top of the flagpole T. [4]
(e) Calculate the shortest distance from C to the line segment AB. [2]
Question
Calculate the length of side BC. [3]
Hint
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Hint
Apply the cosine rule: \(BC = \sqrt{85^2 + 110^2 – 2 \times 85 \times 110 \cos(55°)}\).
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\(BC = \sqrt{85^2 + 110^2 – 2 \times 85 \times 110 \cos(55°)} \approx 92.7\) m
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Question
Calculate the area of the field ABC. [3]
Hint
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Use the area formula: \(Area = \frac{1}{2} \times 85 \times 110 \times \sin(55°)\).
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\(Area = \frac{1}{2} \times 85 \times 110 \times \sin(55°) \approx 3830\) m²
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Question
Calculate the height of the flagpole, BT. [2]
Hint
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In triangle ABT, use \(\tan(15°) = \frac{BT}{85}\).
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\(BT = 85 \tan(15°) \approx 22.8\) m
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Question
Calculate the angle of elevation from C to the top of the flagpole T. [4]
Hint
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In triangle TBC, use the tangent ratio: \(\tan(\theta) = \frac{BT}{BC}\).
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\(\theta = \arctan\left(\frac{22.8}{92.7}\right) \approx 13.8°\)
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Question
Calculate the shortest distance from C to the line segment AB. [2]
Hint
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The perpendicular distance from C to AB is given by \(h = AC \sin(55°)\).
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\(h = 110 \sin(55°) \approx 90.1\) m
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