Circles
Chords and Inscribed Polygon
Grade 11
Question:
<p>Two parallel chords of a circle S have length 10 and 14 and are 6 units apart. If a regular polygon of 12 sides is inscribed in a circle S, then find the area of regular polygon.</p>
Step-by-Step Solution
Key Concept: First find the radius of circle S using the two parallel chords and their distance apart (use perpendicular from center to chords). Then calculate the area of a regular 12-gon inscribed in that circle using the formula: Area = (1/2) × n × r² × sin(2π/n).
<p><strong>Step 1: Find radius using parallel chords</strong></p><p>Let O be center, r = radius. For a chord of length 2a at distance d from center: d² + a² = r²</p><p>For chord of length 10: d₁² + 5² = r² → d₁² + 25 = r²</p><p>For chord of length 14: d₂² + 7² = r² → d₂² + 49 = r²</p><p><strong>Step 2: Use the constraint that chords are 6 units apart</strong></p><p>Since chords are on opposite sides of center: d₁ + d₂ = 6</p><p>From equations above: d₁² + 25 = d₂² + 49</p><p>→ d₁² - d₂² = 24</p><p>→ (d₁ - d₂)(d₁ + d₂) = 24</p><p>→ (d₁ - d₂)(6) = 24</p><p>→ d₁ - d₂ = 4</p><p><strong>Step 3: Solve for d₁ and d₂</strong></p><p>d₁ + d₂ = 6 and d₁ - d₂ = 4</p><p>→ d₁ = 5, d₂ = 1</p><p><strong>Step 4: Find radius</strong></p><p>r² = d₁² + 25 = 25 + 25 = 50</p><p>→ r = 5√2</p><p><strong>Step 5: Find area of regular 12-gon</strong></p><p>Area = (1/2) × n × r² × sin(2π/n) = (1/2) × 12 × 50 × sin(π/6)</p><p>= 6 × 50 × (1/2) = 150</p><p>∴ <strong>Answer: 150 square units</strong></p>
Correct Answer: 150