2. [Maximum mark: 15]
A triangular field ABC has side AB = 85 m and side BC = 110 m. The angle ABC is 70°.
(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.
[VISUALIZATION: A 3D diagram showing horizontal triangle ABC with a vertical line segment BT rising from point B. Angle ABC is 70°. AB = 85 m, BC = 110 m, BT = 18 m.]
(c) Calculate the angle of elevation of T from A. [4]
(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]
Question
Find the length of side AC.
Hint
Hint
Hint
Apply the cosine rule: \(AC^2 = AB^2 + BC^2 – 2\times AB\times BC\cos(70°)\).
-
Answer
Answer
-
Mark Scheme
Mark Scheme
-
Explanation
Explanation
Complete Answer
\(AC \approx 114\,m\)
-
Mark Scheme
Mark Scheme
-
Answer Explanation
Answer Explanation
Question
Find the area of the field ABC.
Hint
Hint
Hint
Use the formula for the area of a triangle: \(Area = 0.5 \times AB \times BC \times \sin(70°)\).
-
Answer
Answer
-
Mark Scheme
Mark Scheme
-
Explanation
Explanation
Complete Answer
\(Area \approx 4390\,m^2\)
-
Mark Scheme
Mark Scheme
-
Answer Explanation
Answer Explanation
Question
Calculate the angle of elevation of T from A.
Hint
Hint
Hint
In right triangle ABT, use \(\tan(\theta)=\frac{BT}{AB}\).
-
Answer
Answer
-
Mark Scheme
Mark Scheme
-
Explanation
Explanation
Complete Answer
\(\theta \approx 12.0°\)
-
Mark Scheme
Mark Scheme
-
Answer Explanation
Answer Explanation
Question
Calculate the straight-line distance from C to T.
Hint
Hint
Hint
Use the Pythagorean theorem in right triangle CBT where \(CT = \sqrt{BC^2 + BT^2}\).
-
Answer
Answer
-
Mark Scheme
Mark Scheme
-
Explanation
Explanation
Complete Answer
\(CT \approx 111\,m\)
-
Mark Scheme
Mark Scheme
-
Answer Explanation
Answer Explanation
Question
Point D lies on AC such that BD is perpendicular to AC. Calculate the length BD.
Hint
Hint
Hint
Use the area of triangle ABC expressed as \(\text{Area} = \frac{1}{2}\times AC\times BD\) and solve for BD.
-
Answer
Answer
-
Mark Scheme
Mark Scheme
-
Explanation
Explanation
Complete Answer
\(BD \approx 77.3\,m\)
-
Mark Scheme
Mark Scheme
-
Answer Explanation
Answer Explanation
