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2007 AMC 12A Problem 8

Problem 8 of 25EasierGeometry

A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords are drawn from 1212 to 5,5, from 55 to 10,10, from 1010 to 3,3, and so on, ending back at 12.12. What is the degree measure of the angle at each vertex in the star-polygon?

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Solution

Consider the two chords meeting at the number 5.5. They run to 1212 and to 10,10, so the arc they subtend extends from 1010 to 12.12. That arc spans two of the twelve hour-marks, so its measure is 212⋅360∘=60∘.\tfrac{2}{12}\cdot 360^\circ=60^\circ. By the Inscribed Angle Theorem, the vertex angle is half the arc, or 12⋅60∘=30∘.\tfrac12\cdot 60^\circ=30^\circ. By symmetry every vertex angle equals 30∘.30^\circ. Thus, the correct answer is C.

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Concepts: inscribed angle · arc

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.