Question

圆心为 $O$ 的圆周被分成 $12$ 段相等的弧，并如图标上字母 $A$ 到 $L$ 。角 $x$ 与 $y$ 的和是多少度？

The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y?$

$75$

$80$

$90$

$120$

$150$

答案：C

知识点：弧等腰三角形导角

难度评级：1220

解答：

$12$ 段弧等分圆周，所以每段弧对应的圆心角为 $360^{\circ} / 12 = 30^{\circ}$ 。

$\angle AOE$ 跨过 $4$ 段弧，所以 $\angle AOE = 4 \cdot 30^{\circ} = 120^{\circ}.$ 类似地， $\angle GOI = 2 \cdot 30^{\circ} = 60^{\circ}.$ 两个三角形都是等腰三角形，因此 $x = \dfrac{180 - 120}{2} = 30^{\circ}$ $y = \dfrac{180 - 60}{2} = 60^{\circ}.$ 所以 $x + y = 90^{\circ}$ 。

所以正确答案是 C。

Note that each of the $12$ arcs splits the circle evenly, so they each cover $360^{\circ} / 12 = 30^{\circ}.$

$\angle AOE$ spans $4$ of these arcs, so $\angle AOE = 4 \cdot 30^{\circ} = 120^{\circ}.$ Similarly, $\angle GOI = 2 \cdot 30^{\circ} = 60^{\circ}.$ We also know that both triangles are isosceles since two of their sides are radii. Therefore, $x = \dfrac{180 - 120}{2} = 30^{\circ}$ and $y = \dfrac{180 - 60}{2} = 60^{\circ}.$ Therefore, $x + y = 90^{\circ}.$

Thus, C is the correct answer.

2014 AMC 8 第 15 题

答案：C

难度评级：1220

解答：

其他年份的第 15 题