2024 AMC 8 Problems/Problem 20

1 Problem
2 Solution 1
3 Solution 2
4 Video Solution by Power Solve
5 Video Solution 1 by Math-X (First understand the problem!!!)
6 Video Solution by NiuniuMaths (Easy to understand!)
7 Video Solution 2 by OmegaLearn.org
8 Video Solution 3 by SpreadTheMathLove
9 Video Solution by CosineMethod [🔥Fast and Easy🔥]
10 Video Solution by Interstigation
11 See Also

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?

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

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

Solution 1

The only equilateral triangles that can be formed are through the diagonals of the faces of the square. 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 $\boxed{\textbf{(D) }3}$ ~Math645 ~andliu766 ~e___

Solution 2

Each other compatible point must be an even number of edges away from P, so the compatible points are R, V, and T. Therefore, we must choose two of the three points, because P must be a point in the triangle. So, the answer is ${3 \choose 2} = \boxed{\textbf{(D) }3}$

-ILoveMath31415926535

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/BaE00H2SHQM?si=QSxNpXGLosdIpffx&t=5954

~Math-X

Video Solution by NiuniuMaths (Easy to understand!)

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

~NiuniuMaths

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

Video Solution by Interstigation

https://youtu.be/ktzijuZtDas&t=2353

2024 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

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