2024 AMC 8 Problems/Problem 20

Problem

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?

$\textbf{(A)}0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }6$

Solution 1

The only equilateral triangles that can be formed are through the diagonals of the faces of the square with length $\sqrt{2}$ . From P you have $3$ possible vertices that are possible to form a diagonal through one of the faces. So there are $3$ possible triangles. So the answer $\boxed{\textbf{(D) }3}$ ~Math 645