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Triangular Cubes (Posted on 2004-12-04) Difficulty: 2 of 5

Three points have been chosen randomly from the vertices of a cube. What is the probability that they form (a) an acute triangle; (b) a right triangle?

  Submitted by Erik O.    
Rating: 3.0000 (2 votes)
Solution: (Hide)

The first step in solving this problem is to figure out how many different triangles can be made using the vertices of a cube. Since there are 8 vertices, the first point can be picked from one of 8 vertices, the second point can be picked from the 7 remaining vertices, and the last point of the triangle can be picked from the 6 remaining vertices. There are, therefore, 8𡥚 = 336 ways to draw a triangle on (or inside) a cube, when you disregard the sequence of choosing points for the triangles. Taking the point sequence into account, we can divide 336 by 6 to come up with 56 unique triangles.

Of those 56 triangles, only 8 are regular triangles. To draw a regular triangle in a cube, the first two points must fall on opposing diagonals of a single face. The last point is taken from one of the two points on the opposite face on a corner that does not share an edge with the first two points.

The remaining triangles are all right triangles. Therefore the probability of getting a regular triangle is 8/56 = 1/7 = 0.14286..., and the probability of getting a right triangle is 48/56 = 6/7 = 0.85714...

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some ThoughtsTriangulars cubes REVISITEDAdy TZIDON2010-08-16 14:02:38
AnswerK Sengupta2008-03-15 03:42:16
re(2): i c it this wayAdy TZIDON2004-12-07 09:41:36
re: Same answer, different approachowl2004-12-06 15:01:01
SolutionSame answer, different approachnikki2004-12-06 14:16:01
SolutionThe way I did it (statistical)Larry2004-12-06 03:23:49
re: i c it this wayCharlie2004-12-05 15:15:44
Solutioni c it this wayAdy TZIDON2004-12-05 07:53:06
QuestionGeometry... It's been too longDustin2004-12-05 02:58:48
SolutionSteve Herman2004-12-04 17:25:13
SolutionSolutionBractals2004-12-04 17:21:28
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