AMC12 2007 A

AMC12 2007 A · Q8

AMC12 2007 A · Q8. It mainly tests Circle theorems, Trigonometry (basic).

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?

在钟面上通过从每个数字向顺时针方向数的第五个数字画弦来绘制一个星形多边形。也就是说，画从 12 到 5 的弦，从 5 到 10 的弦，从 10 到 3 的弦，依此类推，最后回到 12。该星形多边形每个顶点处的角的度数是多少？

(A) 20 20

(B) 24 24

(D) 36 36

(E) 60 60

Answer

Correct choice: (C)

正确答案：（C）

Solution

Error creating thumbnail: Unable to save thumbnail to destination We look at the angle between 12, 5, and 10. It subtends $\frac 16$ of the circle, or $60$ degrees (or you can see that the arc is $\frac 23$ of the right angle). Thus, the angle at each vertex is an inscribed angle subtending $60$ degrees, making the answer $\frac 1260 = 30^{\circ} \Longrightarrow \mathrm{(C)}$

考虑由 12、5、10 所成的角。它所对的弧占圆的 $\frac 16$，即 $60$ 度（或者可看出该弧是直角的 $\frac 23$）。因此，每个顶点处的角都是对着 $60$ 度弧的内接角，所以角度为 $\frac 1260 = 30^{\circ} \Longrightarrow \mathrm{(C)}$

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