Triangular Field with Flagpole

00:00 15 marks

A triangular field ABC has side \(AB = 85\text{ m}\) and side \(BC = 110\text{ m}\). The angle \(ABC\) is \(70^\circ\).

(a) Find the length of side AC. [3]

(b) Find the area of the field ABC. [3]

A vertical flagpole, BT, of height 18 m is located at corner B, with T being the top of the flagpole. The ground is horizontal.

(d) Calculate the straight-line distance from C to T. [3]

(e) Point D lies on AC such that BD is perpendicular to AC. Calculate the length BD. [2]

Part a

3 minutes 3 marks

Question

Find the length of side AC. [3]

Hint Hint

Hint

Use the cosine rule: \(AC^2 = AB^2 + BC^2 – 2 \cdot AB \cdot BC \cdot \cos(\angle ABC)\).

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AC = 114 m

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Part b

3 minutes 3 marks

Question

Find the area of the field ABC. [3]

Hint Hint

Hint

Use \(\text{Area}=\tfrac12\,AB\times BC\times\sin(\angle ABC)\).

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Area = 4390 m²

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Part c

4 minutes 4 marks

Question

Calculate the angle of elevation of T from A. [4]

Hint Hint

Hint

In right triangle ABT, use \(\tan(\angle BAT)=BT/AB\).

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Angle of elevation = 12.0°

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Part d

3 minutes 3 marks

Question

Calculate the straight-line distance from C to T. [3]

Hint Hint

Hint

In right triangle CBT, use \(CT=\sqrt{BC^2+BT^2}\).

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CT = 111 m

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Part e

2 minutes 2 marks

Question

Point D lies on AC such that BD is perpendicular to AC. Calculate the length BD. [2]

Hint Hint

Hint

Use the area of triangle ABC: Area = ½·AC·BD, then solve for BD.

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BD = 77.3 m

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