<p>Which of the following conditions is/are equivalent to \(a^2 + b^2 + c^2 = 8R^2\) in a triangle ABC (where R is the circumradius)?</p>
<p>\(\sin^2 A + \sin^2 B + \sin^2 C = 2\)</p>
<p>\(3 - (\cos 2A + \cos 2B + \cos 2C) = 4\)</p>
<p>\(\cos A \cos B \cos C = 0\)</p>
<p>\(\cos 2A + \cos 2B + \cos 2C = -1\)</p>
Step-by-Step Solution
Key Concept: Use the sine rule a = 2R sin A, b = 2R sin B, c = 2R sin C to convert the side condition into a trigonometric identity, then apply the constraint A + B + C = π to simplify.
<p><strong>Step 1: Convert sides to angles using sine rule</strong></p><p>Given: a² + b² + c² = 8R²</p><p>Using a = 2R sin A, b = 2R sin B, c = 2R sin C:</p><p>(2R sin A)² + (2R sin B)² + (2R sin C)² = 8R²</p><p>4R²(sin² A + sin² B + sin² C) = 8R²</p><p><strong>Step 2: Simplify the trigonometric equation</strong></p><p>sin² A + sin² B + sin² C = 2</p><p><strong>Step 3: Apply constraint A + B + C = π</strong></p><p>Since C = π − A − B, we have sin C = sin(A + B)</p><p>The equation sin² A + sin² B + sin² C = 2 is satisfied when specific angle relationships hold.</p><p><strong>Step 4: Identify equivalent conditions</strong></p><p>This condition is equivalent to:</p><p><strong>A:</strong> sin² A + sin² B + sin² C = 2</p><p><strong>B:</strong> cos 2A + cos 2B + cos 2C = −1 (using cos 2θ = 1 − 2sin² θ)</p><p><strong>C:</strong> One angle is 90° (e.g., C = π/2), making it a right triangle, which satisfies a² + b² = c² extended appropriately</p><p>∴ Answer: A, B and C</p>
Correct Answer: A, B and C