1958 AMC 12 第 15 题

Question

一个四边形内接于圆。在四边形外部的四个弓形中各作一个圆周角，则这四个角的度数之和为：

A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is:

1080

900

720

540

360

答案：D

知识点：圆内接四边形圆周角弧角度和

难度评级：1630

小提示：

四边形各边所截的四段弧的度数之和为

360^\circ

Let the four arcs cut off by the quadrilateral’s sides have measures summing to

360^\circ

大提示：

位于某边外侧弓形内的圆周角所对的是互补的优弧

An angle in the outside segment of a side subtends the complementary major arc

解答：

设各边所截的四段劣弧度数为

\alpha_1,\ldots,\alpha_4

，其和为

360^\circ

。与弧

\alpha_i

对应的外侧弓形内的圆周角所对的是另一段弧，因此其度数为

\frac{360^\circ-\alpha_i}{2}\text{。}

所以四个角的度数之和为

\frac{4(360^\circ)-360^\circ}{2}=540^\circ\text{。}

所以正确答案为 D。

Let the four minor arcs cut off by the sides have measures

\alpha_1,\ldots,\alpha_4,

whose sum is

360^\circ.

The angle in the outside segment corresponding to arc

\alpha_i

subtends the other arc, so its measure is

\frac{360^\circ-\alpha_i}{2}.

The sum of the four angles is therefore

\frac{4(360^\circ)-360^\circ}{2}=540^\circ.

Thus, the correct answer is D.

1958 AMC 12 第 15 题

答案：D

难度评级：1630

小提示：

大提示：

解答：

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