atalpanditgmat wrote:

What is the probability of randomly selecting one of the shortest diagonals from among all the diagonals of a regular octagon?

Though i can find number of diagonals using n(n-3)/2, I have no idea what is shortest diagonal of regular octagon?

Shed some light...

Look at the diagram. The diagonals will be of varying lengths depending on which two points you join. When you join adjacent points, you get the sides. When you join points with one point between them, you get the shortest diagonal. When you join points further off, you get longer diagonals. So we need to figure the number of shortest diagonals.

Attachment:

Octagon.jpg [ 8.62 KiB | Viewed 1289 times ]
Each point will join with two such points to give 2 diagonals e.g. the red diagonals and the green diagonals. So there will be a total of 8*2 = 16 such diagonals. But notice that you have counted each diagonal twice (for both the points). Hence the number of shortest diagonals will be 16/2 = 8.

Required Probability = 8/20 = 2/5

Note that total number of diagonals can be calculated as nC2 - n. Put n = 8 here to get 20.

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