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Difference between revisions of "2024 AMC 8 Problems/Problem 20"

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Any three vertices of the cube <math>PQRSTUVW</math>, shown in the figure below, can be connected to form a triangle. (For example, vertices <math>P</math>, <math>Q</math>, and <math>R</math> can be connected to form isosceles <math>\triangle PQR</math>.) How many of these triangles are equilateral and contain <math>P</math> as a vertex?
 
Any three vertices of the cube <math>PQRSTUVW</math>, shown in the figure below, can be connected to form a triangle. (For example, vertices <math>P</math>, <math>Q</math>, and <math>R</math> can be connected to form isosceles <math>\triangle PQR</math>.) How many of these triangles are equilateral and contain <math>P</math> as a vertex?
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[asy]
 
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unitsize(4);
 
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pair P,Q,R,S,T,U,V,W;
 
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P=(0,30); Q=(30,30); R=(40,40); S=(10,40); T=(10,10); U=(40,10); V=(30,0); W=(0,0);
 
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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);
 
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dot(P);
 
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dot(Q);
 
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dot(R);
 
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dot(S);
 
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dot(T);
 
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dot(U);
 
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dot(V);
 
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dot(W);
 
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label("<math>P</math>",P,NW);
 
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label("<math>Q</math>",Q,NW);
 
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label("<math>R</math>",R,NE);
 
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label("<math>S</math>",S,N);
 
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label("<math>T</math>",T,NE);
 
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label("<math>U</math>",U,NE);
 
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label("<math>V</math>",V,SE);
 
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label("<math>W</math>",W,SW);
 
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[/asy]
 
  
 
<math>\textbf{(A)}0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }6</math>
 
<math>\textbf{(A)}0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }6</math>

Revision as of 00:43, 28 January 2024

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

~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

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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