Quandaries
and Queries 

I have a problem for a 10th grader? Question 1. Question
2. Thanks, 

Hi Glenn, I am going to assume that you are dealing with a regular octagon. First you need to know how many diagonals a regular octagon has and then how many have the longest length. Draw an octagon, select one vertex and construct each diagonal from this vertex.You will see there are 5 such diagonals. Thus for each of the 8 vertices you can draw 5 diagonals and hence you have constructed 5 8 = 40 diagonals. But you have constructed each diagonal twice, once from each of its ends. Thus there are 20 diagonals in a regular octagon. A look at the diagram above shows that one of the diagonals constructed is longer than the others. Thus you construct one long diagonal from each vertex, and hence 8 long diagonals. Again you have constructed each one twice so there are 4 long diagonals. Thus if you select one diagonal from all the diagonals in a regular octagon there are 4 chances in 20 that you will select one of the longest ones, and hence a probability of ^{4}/_{20} = ^{1}/_{5}. What about the shortest? Andrei and Penny


