Question

圆心为 $O$ ， $\overline{AB}$ 是直径，点 $C$ 在圆上，且 $\angle COB = 50^\circ$ 。 $\angle CAB$ 的度数是多少？

A circle is centered at $O,$ $\overline{AB}$ is a diameter and $C$ is a point on the circle with $\angle COB = 50^\circ.$ What is the degree measure of $\angle CAB?$

$20$

$25$

$45$

$50$

$65$

答案：B

知识点：圆等腰三角形导角

难度评级：960

解答：

参考下图：

由于直径的两端与圆心在同一直线上， $\angle AOC = 180^{\circ} - 50^{\circ} = 130^{\circ}.$

$\triangle AOC$ 是等腰三角形，所以 $\angle CAO = \dfrac{180^{\circ} - 130^{\circ}}{2} = 25^{\circ}.$

因为 $\angle CAB = \angle CAO$ ，所以 $\angle CAB = 25^{\circ}$ 。

所以正确答案是 B。

Consider the following diagram:

We have that $\angle AOC = 180^{\circ} - 50^{\circ} = 130^{\circ}.$

Since $\triangle AOC$ is isosceles, we have that $\angle CAO = \dfrac{180^{\circ} - 130^{\circ}}{2} = 25^{\circ}.$

Since $\angle CAB = \angle CAO,$ we have that $\angle CAB = 25^{\circ}.$

Thus, B is the correct answer.

2010 AMC 10B 第 6 题

答案：B

难度评级：960

解答：

其他年份的第 6 题