A point p in the interior of a regular hexagon is at distance 8,8,16 units from three consecutive vertices of the hexagon, respectively. If r is radius of the circumscribed circle of the hexagon, what is the integer closest to r
Answers
Answer:
integer closest to r = 14
Step-by-step explanation:
A point p in the interior of a regular hexagon is at distance 8,8,16 units from three consecutive vertices of the hexagon, respectively. If r is radius of the circumscribed circle of the hexagon, what is the integer closest to r
Lets call vertices A , B & C
as AP = BP = 8 unit
=> ∠BAP = ∠ABP = α ( let say)
⇒ ∠APB = 180° - α - α = 180°-2α
AB² = AP² + BP² -2AB.BPCos(180°-2α)
=> AB² = 8² + 8² - 2*8*8Cos(180°-2α)
=> AB² = 2*8² + 2*8²Cos(2α)
=> AB² = 2*8²(1 + Cos2α)
=> AB² = 2*8²(2Cos²α)
=> AB = 16Cosα
∠ABC = 120° ( Hexagon angles)
=> ∠PBC = 120° - α
=> CP² = BP² + BC² -2BP*BCCos(120° - α)
BC = AB
=> 16² = 8² + (16Cosα)² - 2*8*16Cosα*Cos(120° - α)
=> 4 = 1 + 4Cos²α - 4Cosα (Cos120°Cosα + Sin120°Sinα)
=> 3 = 4Cos²α - 4Cosα(-Cosα/ 2 + (√3/2) Sinα)
=> 3 = 6Cos²α - 2√3Cosα Sinα
=> 3 = 3(1+Cos2α) - √3Sin2α
=> √3Sin2α =3*Cos2α
=> Tan2α = √3
=> 2α = 60°
=> α = 30°
AB = 16Cosα = 16 * Cos30° = 16 *√3/2 = 8√3 = 13.856
≈ 14
in Hexagon Radius = Side of Hexagon
=> integer closest to r = 14