Question

一个四边形内接于圆。在四边形各边所截出的四段弧上分别作圆周角，且这些角不与各顶点之间的边相交，则四个角之和为：

A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, without intersecting the sides between vertices, their sum will be:

$180^\circ$

$540^\circ$

$360^\circ$

$450^\circ$

$1080^\circ$

答案：B

知识点：圆周角弧角度和

难度评级：1630

小提示：

设四段连续圆弧的度数为 $a,b,c,d$

Let the four consecutive arc measures be $a,b,c,d$

大提示：

顶点位于度数为 $a$ 的圆弧上的圆周角截得另外三段弧，其总度数为 $360^\circ-a$

An angle whose vertex lies in the arc of measure $a$ intercepts the other three arcs, of total measure $360^\circ-a$

解答：

设四段弧的度数为 $a,b,c,d$ ，其和为 $360^\circ$ 。位于度数为 $a$ 的圆弧上的角截得另外三段弧，所以其度数为 $\frac{360^\circ-a}{2}$ ，其余三个角同理。四个角之和为 $\begin{aligned} &\frac{4(360^\circ)-(a+b+c+d)}2\\ &\qquad=\frac{1080^\circ}{2}\\ &\qquad=540^\circ \end{aligned}\text{。}$

因此，正确答案是 B。

Let the four arcs have measures $a,b,c,d,$ whose sum is $360^\circ.$ The angle placed in the arc of measure $a$ intercepts the other three arcs, so it measures $\frac{360^\circ-a}{2},$ and similarly for the others. Their sum is $\begin{aligned} &\frac{4(360^\circ)-(a+b+c+d)}2\\ &\qquad=\frac{1080^\circ}{2}\\ &\qquad=540^\circ. \end{aligned}$

Thus, the correct answer is B.

1954 AMC 12 第 13 题

答案：B

难度评级：1630

小提示：

大提示：

解答：

其他年份的第 13 题