Help with this geometry/probability question?
- PuzzlingLv 71 month agoBest Answer
In order for △ABC and △XYZ to be similar, the corresponding angles must be congruent:
∠A ≅ ∠X
∠B ≅ ∠Y
∠C ≅ ∠Z
Here we assume a scalene triangle. The angles for △XYZ can be arranged in 3! = 6 ways.
So the probability that everything matches for a scalene triangle (and hence they are similar) is:
Here we have two congruent angles in an isosceles triangle. There would be 3! = 6 ways to arrange the angles, but 2 of them would be identical, so divide by 2. That means there are 3 ways to align the angles with only 1 being right.
So the probability that everything matches for an isosceles triangle (and hence they are similar) is:
In an equilateral triangle, all the angles are the same (60°), so any arrangement will automatically match.
The probability that everything matches for an equilateral triangle is: