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Hexagonal Dilemma (Posted on 2004-06-02) Difficulty: 4 of 5
A hexagon with sides of length 2, 7, 2, 11, 7, 11 is inscribed in a circle. Find the radius of the circle.

As suggested, *if* it matters, you may assume that the sides listed are given in order

No Solution Yet Submitted by SilverKnight    
Rating: 4.0000 (5 votes)

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Solution Geometric solution | Comment 17 of 18 |

The sides {7, 2, 11} may be inscribed in a semicircle of the same radius as the original circle.

Let AB = d be the diameter of the semicircle, BC = 11, CD = 2, and AD = 7.  Draw AC of length x, and BD of length y.

Note that angles ADB and ACB are right angles.  Hence, by Pythagoras:
x2 = d2 - 121
y2 = d2 - 49

Ptolemy's Theorem states that in a cyclic quadrilateral the sum of the products of the two pairs of opposite sides equals the product of its two diagonals.
Hence 2d + 77 = xy.

Therefore (2d + 77)2 = (d2 - 121)(d2 - 49).

Simplifying, and dividing by d, we obtain d3 - 174d - 308 = 0.
This factorizes, giving (d - 14)(d2 + 14d + 22) = 0.
The quadratic factor has negative real roots, so d = 14 is the only positive real root.

Therefore the radius of the circle in which the hexagon is inscribed is 7.


  Posted by Nick Hobson on 2005-06-13 14:13:25
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