in a octagon two diagonals are chosen, what is the probability that the two diagonals are parallel

Question

 it is a regular octagon

Answer 1

 

 

I considered REGULAR octagon ABCDEFGH.  (This must be a regular octagon for problem to make sense.)

# of diagonals is 8C2 (pairings of vertices) - 8 (discounts sides) = 20.

The # of distinct pairs of diagonals is 20C2 = 20(19)/2 = 190.

I tabulated all diagonals and what other diagonal(s) are parallel thereto:

AC // HD, EG   3 pairs //

AD // EH          

AE // BD, FH   3 pairs //

AF // BE

AG // CE, BF   3 pairs //

BD  already listed

BE  already listed

BF  already listed

BG // CF

BH // DF

CE see above

CF see above

CG // BH, DF   3 pairs //

CH // DG

DF see above

DG see above

DH see above

EG see above

EH see above.

THE TOTAL # OF possible parallel pairs pairs is 17, unless there's an error in my accounting.

Therefore I say P(randomly selected pair of diagonals are parallel) is 17/190.