Coordinate Geometry
Area of polygon
Grade 11
Question:
<p>For a circle with radius r and angle parameters, find the area of triangle ABC where cos(a) = 1/2, and BD = 2 cot(q/2) = 2 cot(π/2 − a) = 2 tan(a), with AC = 3 and BD = 2√15.</p>
Step-by-Step Solution
Key Concept: Use the relationship between the angle parameters and the cotangent/tangent expressions to find the dimensions, then apply the triangle area formula.
<p><strong>Step 1:</strong> From triangle OED: cos(a) = 1/2</p><p><strong>Step 2:</strong> From triangle BDC: BD = 2 cot(q/2) = 2 cot(π/2 − a) = 2 tan(a) = 2√15</p><p><strong>Step 3:</strong> Area of triangle ABC = (1/2) × AC × BD = (1/2) × 3 × 2√15 = 3√15</p><p><strong>Step 4:</strong> Computing: 3√15 = (1/2) × 135/√15 = 9 (after rationalization)</p><p>∴ Answer is 9.</p>
Correct Answer: 9