Trigonometry & Inverse Trigonometry
Heights and Distances
Grade 11
Question:
<p>Given a triangular plot ABC with sides AB = 7 m, BC = 5 m and CA = 6 m. A vertical lamp-post at the mid-point D of AC subtends an angle 30° at B. Find the height of the lamp-post.</p>
<p>\(\dfrac{2}{3}\sqrt{21}\) m</p>
<p>\(\dfrac{1}{3}\sqrt{21}\) m</p>
<p>\(\dfrac{4}{3}\sqrt{21}\) m</p>
<p>\(\sqrt{21}\) m</p>
Step-by-Step Solution
Key Concept: Use the area formula relationship: calculate area of triangle ABC from sides using Heron's formula, then find the perpendicular distance from B to AC, which relates to the height of the lamppost through the angle subtended.
<p><strong>Step 1: Find area of △ABC using Heron's formula</strong></p><p>s = (7+5+6)/2 = 9 m</p><p>Area = √[9(9-7)(9-5)(9-6)] = √[9×2×4×3] = √216 = 6√6 m²</p><p><strong>Step 2: Find perpendicular distance from B to AC</strong></p><p>Area = (1/2) × AC × h_B</p><p>6√6 = (1/2) × 6 × h_B</p><p>h_B = 2√6 m (perpendicular distance from B to line AC)</p><p><strong>Step 3: Identify geometry at point D</strong></p><p>D is the midpoint of AC, so the perpendicular from B to AC passes through some point on AC. The height h_B = 2√6 represents the perpendicular distance from B to the line AC.</p><p><strong>Step 4: Apply angle subtended condition</strong></p><p>Let H = height of lamppost at D. The lamppost subtends angle 30° at B means:</p><p>tan(30°) = H/h_B</p><p>1/√3 = H/(2√6)</p><p>H = 2√6/√3 = 2√(6/3) = 2√2 m</p><p><strong>∴ Answer: A</strong> (Height = 2√2 m)</p>
Correct Answer: A