Hexagon!
Geometry
Level
3
A hexagon with consecutive sides of lengths 2,2,7,7,11 and 11 is inscribed in a circle. Find the radius of the circle.
The answer is 7.
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1 solution
Consider half of the circle, with the quadrilateral ABCD, AD being the diameter. AB = 2, BC = 7, and CD = 11. Construct diagonals AC and BD.
Notice that these diagonals form right triangles. You get the following system of equations:
(AC)(BD) = 7(AD) + 22 from (Ptolemy's Theorem)
(AC)^2 = (AD)^2 - 121
(BD)^2 = (AD)^2 - 4
Solving these equations
Gives AD = 14. So the radius is 7.