Question

在一个等边三角形中，每个内角都由两条射线三等分。每个顶点处中间那个 $20^\circ$ 角的内部相交部分，是一个凸六边形的内部。这个六边形最小内角的度数是多少？

In an equilateral triangle each interior angle is trisected by a pair of rays. The intersection of the interiors of the middle $20^\circ$ -angle at each vertex is the interior of a convex hexagon. What is the degree measure of the smallest angle of this hexagon?

$80$

$90$

$100$

$110$

$120$

答案：C

知识点：等边三角形导角角度和

难度评级：1310

解答：

设等边三角形为 $ABC$ 。每个 $60^\circ$ 角被分成三个 $20^\circ$ 部分。取从 $A$ 和 $B$ 出发的外侧三等分线：它们形成的三角形底角都是 $\tfrac23 \cdot 60^\circ = 40^\circ$ ，所以对应的六边形顶角为 $180^\circ - 2\cdot 40^\circ = 100^\circ$ 。从 $A$ 和 $B$ 出发的内侧三等分线形成底角为 $20^\circ$ 的三角形，顶角为 $180^\circ - 2\cdot 20^\circ = 140^\circ$ ，由对顶角可得到相对的六边形角。因此六个角交替为 $100^\circ$ 和 $140^\circ$ 。最小角为 $100^\circ$ 。因此正确答案是 C。

Label the equilateral triangle $ABC.$ Each $60^\circ$ angle splits into three $20^\circ$ pieces. Take the outermost trisectors from $A$ and $B$ : they meet at base angles $\tfrac23 \cdot 60^\circ = 40^\circ,$ so the hexagon vertex there has angle $180^\circ - 2\cdot 40^\circ = 100^\circ.$ The innermost trisectors from $A$ and $B$ meet at base angles $20^\circ,$ giving apex $180^\circ - 2\cdot 20^\circ = 140^\circ,$ and by vertical angles that's the opposite hexagon angle. So the six angles alternate $100^\circ$ and $140^\circ.$ The smallest is $100^\circ.$ Therefore, the answer is C.

2025 AMC 10A 第 6 题

答案：C

难度评级：1310

解答：

其他年份的第 6 题