2022 · Q68
In a triangle ABC, ∠A = 2θ, ∠B = ∠C = 4θ and θ satisfies 4sin²θ + 2sinθ - 1 = 0. What is the ratio of BC to AB?
Elementary MathematicsGeometry: Lines, Angles and Triangles
- A5 - 1
- B(5-1)2
- C(5-1)4
- D2(5 - 1)
Explanation
2θ+4θ+4θ = 180° gives θ = 18° (consistent with 4sin²θ+2sinθ−1 = 0). By the sine rule, BC/AB = sin2θ/sin4θ = 1/(2cos36°) = 2/(√5+1) = (√5−1)/2. The keyed option is B.