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?

(In reply to i c it this way by Ady TZIDON)

Right angled triangles can be made not only restricted to one side, but also using as legs one edge and a diagonal of an adjacent side to the two sides that form that edge (not a diagonal of one of the two sides themselves).  The hypotenuse is then a main diagonal of the cube. The sides are in ratio 1:sqrt(2):sqrt(3)

There are 12 edges, and two choices of diagonal for each, so another 24/56 or 3/7 of the cases are like this, leaving only 1/7 for acute triangles.

 

Posted by Charlie on 2004-12-05 15:15:44