If and be a regular hexagon inscribed in a circle of unit radius. Then, the product of the lengths of the line segments , and is
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: C
Solution
The vertices of a regular hexagon inscribed in a unit circle are separated by 60° each. The chord length between vertices k steps apart is 2sin(kπ/6).
- A₀A₁: 1 step apart → 2sin(π/6) = 2·(1/2) = 1
- A₀A₂: 2 steps apart → 2sin(2π/6) = 2sin(π/3) = √3
- A₀A₄: 4 steps apart (equivalently 2 steps the other way) → 2sin(4π/6) = 2sin(2π/3) = √3
Product = 1 · √3 · √3 = 3.
- A₀A₁: 1 step apart → 2sin(π/6) = 2·(1/2) = 1
- A₀A₂: 2 steps apart → 2sin(2π/6) = 2sin(π/3) = √3
- A₀A₄: 4 steps apart (equivalently 2 steps the other way) → 2sin(4π/6) = 2sin(2π/3) = √3
Product = 1 · √3 · √3 = 3.
[1998]