Question

在钟面上画一个星形多边形：从每个数字向顺时针数第五个数字画一条弦。也就是说，画从 $12$ 到 $5$ 、从 $5$ 到 $10$ 、从 $10$ 到 $3$ 的弦，依此类推，最后回到 $12$ 。这个星形多边形每个顶点处的角是多少度？

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?

$20$

$24$

$30$

$36$

$60$

答案：C

知识点：圆周角弧

难度评级：1350

解答：

考虑在数字 $5$ 处相交的两条弦。它们分别连到 $12$ 和 $10$ ，因此它们所对的弧从 $10$ 到 $12$ 。

这段弧跨过十二个小时刻度中的两个，所以其度数为 $\tfrac{2}{12}\cdot 360^\circ=60^\circ$ 。

根据圆周角定理，顶点角等于所对弧的一半，即 $\tfrac12\cdot 60^\circ=30^\circ$ 。由对称性，每个顶点角都是 $30^\circ$ 。

因此，正确答案是 C。

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.

2007 AMC 12A 第 8 题

答案：C

难度评级：1350

解答：

其他年份的第 8 题