Question

下图所示立方体 $PQRSTUVW$ 的任意三个顶点都可以连接成一个三角形。（例如，顶点 $P$ 、 $Q$ 和 $R$ 可以连成等腰三角形 $\bigtriangleup PQR$ 。）这些三角形中，有多少个是等边三角形并且包含顶点 $P$ ？

Any three vertices of the cube $PQRSTUVW,$ shown in the figure below, can be connected to form a triangle. (For example, vertices $P,$ $Q,$ and $R$ can be connected to form isosceles $\bigtriangleup PQR.$ ) How many of these triangles are equilateral and contain $P$ as a vertex?

$0$

$1$

$2$

$3$

$6$

答案：D

知识点：正方体等边三角形

难度评级：1420

解答：

为了得到等边三角形，三条边都必须等于立方体的面对角线。逐一检查可得 $\triangle PVT$ 、 $\triangle PRT$ 和 $\triangle PRV$ 。

共有三个。正确答案是 D。

We first note that we can only form equilateral triangles if we go through the diagonals of the square faces, otherwise at least one angle of the triangle will be different. Afterwards, it is easy to exhaust all possible equilateral triangles that can be formed: $\triangle PVT,$ $\triangle PRT,$ and $\triangle PRV.$

Thus, D is the correct answer.

2024 AMC 8 第 20 题

答案：D

难度评级：1420

解答：

其他年份的第 20 题