AMC8 2014

AMC8 2014 · Q15

AMC8 2014 · Q15. It mainly tests Circle theorems.

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 $y$?

圆心为 $O$ 的圆的圆周被分成了 $12$ 个相等的弧段，从图中所示的字母 $A$ 到 $L$。角度 $x$ 和 $y$ 的和是多少度？

(A) 75 75

(B) 80 80

(D) 120 120

(E) 150 150

Answer

Correct choice: (C)

正确答案：（C）

Solution

The measure of an inscribed angle is half the measure of its corresponding central angle. Since each unit arc is $\frac{1}{12}$ of the circle's circumference, each unit central angle measures $\frac{360}{12}^{\circ}=30^{\circ}$. From this, $\angle EOG = 60^{\circ}$, so $x = 30^{\circ}$. Also, $\angle AOI = 120^{\circ}$, so $y = 60^{\circ}$. The number of degrees in the sum of both angles is $30 + 60 = \boxed{(C)\ 90}.$

圆周角的度数是对应圆心角度数的一半。由于每个弧段是整个圆周的 $\frac{1}{12}$，所以每个对应的圆心角度数为 $\frac{360}{12}^{\circ} = 30^{\circ}$。因此，$\angle EOG = 60^{\circ}$，所以 $x = 30^{\circ}$。另外，$\angle AOI = 120^{\circ}$，所以 $y = 60^{\circ}$。两个角度的和是 $30 + 60 = \boxed{(C)\ 90}$。

Topics

GE.006 Circle theorems