1954 AMC 12 第 13 题

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