2007 AMC 12A Problem 8

Question

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

12

to

5,

from

5

to

10,

from

10

to

3,

and so on, ending back at

12.

What is the degree measure of the angle at each vertex in the star-polygon?

20

24

30

36

60

Answer: C

Concepts:inscribed angle arc

Difficulty rating: 1350

Small hint:

Each vertex angle is an inscribed angle in the circle through the clock numbers

Big hint:

The two chords meeting at a vertex cut off an arc spanning two hour-marks, or

60^\circ

Solution:

Consider the two chords meeting at the number

5.

They run to

12

and to

10,

so the arc they subtend extends from

10

to

12.

That arc spans two of the twelve hour-marks, so its measure is

\tfrac{2}{12}\cdot 360^\circ=60^\circ.

By the Inscribed Angle Theorem, the vertex angle is half the arc, or

\tfrac12\cdot 60^\circ=30^\circ.

By symmetry every vertex angle equals

30^\circ.

Thus, the correct answer is C.