Question

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?

$0$

$1$

$2$

$3$

$6$

Solution:

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 Problem 20

Solution:

Problem 20 in Other Years