Question

两个全等圆分别以点 $A$ 和 $B$ 为圆心，并且每个圆都经过另一个圆的圆心。经过 $A$ 和 $B$ 的直线延长后与两个圆分别交于点 $C$ 和 $D$ 。

两圆相交于两点，其中一点为 $E$ 。 $\angle CED$ 的度数是多少？

Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D.$

The circles intersect at two points, one of which is $E.$ What is the degree measure of $\angle CED?$

$90$

$105$

$120$

$135$

$150$

答案：C

知识点：等边三角形圆周角导角

难度评级：1510

视频讲解：

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文字解答：

因为 $AE = EB = AB$ ，它们都是全等圆的半径，所以它们构成等边三角形， $\angle AEB = 60^{\circ}$ 。

又因为 $\overline{DB}$ 和 $\overline{AC}$ 是直径，所以因此得 $\angle DEB=\angle AEC=90^\circ$ 。 $\begin{aligned} \angle CED &= \angle DEB+\angle AEC\\ &\qquad-\angle AEB\\ &=90^\circ+90^\circ-60^\circ\\ &=120^\circ. \end{aligned}$

所以正确答案是 C。

We know that $AE = EB = AB$ since they are all radii of congruent circles, so they form an equilateral triangle, which means that $\angle AEB = 60^{\circ}.$

The segments $\overline{DB}$ and $\overline{AC}$ are diameters, so $\angle DEB=\angle AEC=90^\circ.$ Therefore, $\begin{aligned} \angle CED &= \angle DEB+\angle AEC\\ &\qquad-\angle AEB\\ &=90^\circ+90^\circ-60^\circ\\ &=120^\circ. \end{aligned}$

Thus, C is the correct answer.

2016 AMC 8 第 23 题

答案：C

难度评级：1510

视频讲解：

文字解答：

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