AMC8 2024

AMC8 2024 · Q20

AMC8 2024 · Q20. It mainly tests Triangles (properties), Polygons.

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 $\triangle PQR$.) How many of these triangles are equilateral and contain $P$ as a vertex?

立方体 $PQRSTUVW$ 的任意三个顶点可以连接形成一个三角形。（例如，顶点 $P$、$Q$ 和 $R$ 可以连接形成等腰 $\triangle PQR$）。其中有多少个这样的三角形是等边三角形且以 $P$ 为顶点？

(A) 0 0

(B) 1 1

(D) 3 3

(E) 6 6

Answer

Correct choice: (D)

正确答案：（D）

Solution

The only equilateral triangles that can be formed are through the diagonals of the faces of the square. From P you have $3$ possible vertices that are possible to form a diagonal through one of the faces. Therefore, there are $3$ possible triangles. So the answer is $\boxed{\textbf{(D) }3}$

唯一能形成的等边三角形是通过正方形面的对角线。从 $P$ 你有 $3$ 个可能的顶点，可以通过其中一个面形成对角线。因此，有 $3$ 个可能的三角形。所以答案是 $\boxed{\textbf{(D) }3}$

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