Problem 18 · 2011 Math Kangaroo
Hard
Geometry & Measurement
The sides AB, BC, CD, DE, EF and FA of a hexagon all touch the same circle. The sides AB, BC, CD, DE and EF, in this order, measure 4, 5, 6, 7 and 8. How long is side FA?
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Answer: D — 6
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Hint 1 of 2
For a polygon whose sides all touch one circle, the two tangent lengths from each vertex are equal.
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Hint 2 of 2
That makes the alternating sums of side lengths equal.
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Approach: equal alternating sums in a tangential polygon
- In a tangential hexagon, AB + CD + EF = BC + DE + FA.
- So 4 + 6 + 8 = 5 + 7 + FA, giving 18 = 12 + FA.
- Thus FA = 6.
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