<p>In a triangle ABC, \(\angle C = 2\angle A\) and \(b = 2a\). Which of the following are correct?</p>
Step-by-Step Solution
Key Concept: Use the sine rule (a/sin A = b/sin B) combined with the angle constraint ∠C = 2∠A to establish relationships between sides and angles, then apply the constraint b = 2a to find specific angle values.
<p><strong>Step 1:</strong> Apply sine rule: a/sin A = b/sin B</p><p>Given b = 2a, we get: a/sin A = 2a/sin B</p><p>Therefore: sin B = 2 sin A ... (i)</p><p><strong>Step 2:</strong> Use angle sum: A + B + C = 180°</p><p>Since C = 2A: A + B + 2A = 180°</p><p>So: B = 180° - 3A ... (ii)</p><p><strong>Step 3:</strong> Substitute (ii) into (i):</p><p>sin(180° - 3A) = 2 sin A</p><p>sin(3A) = 2 sin A</p><p><strong>Step 4:</strong> Expand sin 3A = 3 sin A - 4 sin³ A:</p><p>3 sin A - 4 sin³ A = 2 sin A</p><p>sin A - 4 sin³ A = 0</p><p>sin A(1 - 4 sin² A) = 0</p><p><strong>Step 5:</strong> Since A > 0: sin² A = 1/4, so sin A = 1/2</p><p>This gives A = 30°</p><p><strong>Step 6:</strong> Calculate other angles:</p><p>C = 2A = 60°</p><p>B = 180° - 30° - 60° = 90°</p><p>∴ Triangle ABC is a right-angled triangle with B = 90°, and the specific angle measures are A = 30°, B = 90°, C = 60°</p>
Correct Answer: B and D