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2009 AMC 12A Problem 22

Problem 22 of 25HarderGeometry

A regular octahedron has side length 1.1. A plane parallel to two of its opposite faces cuts the octahedron into two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area abc,\dfrac{a\sqrt{b}}{c}, where a,a, b,b, and cc are positive integers, aa and cc are relatively prime, and bb is not divisible by the square of any prime. What is a+b+c?a + b + c?

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Solution

Let the two parallel faces be triangles. The plane passes through the midpoints of the six edges not on those faces, forming an equilateral hexagon of side 12,\dfrac{1}{2}, which by symmetry is also equiangular and hence regular. A regular hexagon is six equilateral triangles, so its area is 6⋅34(12)2=338.6\cdot\frac{\sqrt{3}}{4}\left(\frac{1}{2}\right)^2 = \frac{3\sqrt{3}}{8}. Thus a=3,a = 3, b=3,b = 3, c=8,c = 8, and a+b+c=14.a + b + c = 14. Thus, the correct answer is E.
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Tagged: 3D geometry · regular polygon · area

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