Olympiad workbookIOQMStraight Lines
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 ?
Correct answer
14
Step-by-step solution
Note that CF =2 AB , PA =2 PC & PB =2 PF Hence PAB is similar to PFC , hence A ₁ P ₁ C & B ₁ P ₁ ~F are collinear. Let each side of hexagon be equal to x . Let, Q & R be foot of altitudes from P to base A B & C F respectively. So R is centre of hexagon Now 1 3 3 x 2 = 64- x^2 4 x^2 12 =64- x^2 4 4 x^2 12 =64 x=8 3 Note that circumradius of a regular hexagon = side of regular hexagon Hence r=8 3 13.856 Hence nearest integer =14