2024 AMC 8 Problems/Problem 20

Problem

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

[asy] unitsize(4); pair P,Q,R,S,T,U,V,W; P=(0,30); Q=(30,30); R=(40,40); S=(10,40); T=(10,10); U=(40,10); V=(30,0); W=(0,0); draw(W--V); draw(V--Q); draw(Q--P); draw(P--W); draw(T--U); draw(U--R); draw(R--S); draw(S--T); draw(W--T); draw(P--S); draw(V--U); draw(Q--R); dot(P); dot(Q); dot(R); dot(S); dot(T); dot(U); dot(V); dot(W); label("",P,NW); label("",Q,NW); label("",R,NE); label("",S,N); label("",T,NE); label("",U,NE); label("",V,SE); label("",W,SW); [/asy]

Solution 1

The only equilateral triangles that can be formed are through the diagonals of the faces of the square with length . From P you have possible vertices that are possible to form a diagonal through one of the faces. So there are possible triangles. So the answer ~Math 645

~andliu766

Video Solution by NiuniuMaths (Easy to understand!)

https://www.youtube.com/watch?v=V-xN8Njd_Lc

~NiuniuMaths

Video Solution by Power Solve

https://www.youtube.com/watch?v=7_reHSQhXv8

Video Solution 1 by Math-X (First understand the problem!!!)

https://youtu.be/N_9qlD9pgL0

~Math-X

Video Solution 2 by OmegaLearn.org

https://youtu.be/m1iXVOLNdlY

Video Solution 3 by SpreadTheMathLove

https://www.youtube.com/watch?v=Svibu3nKB7E

Video Solution by CosineMethod [🔥Fast and Easy🔥]

https://www.youtube.com/watch?v=Xg-1CWhraIM

See Also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.