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What is the probability of randomly selecting one of......

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15 Apr 2013, 21:50
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...
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Re: What is the probability of randomly selecting one of...... [#permalink]

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15 Apr 2013, 22:01
An octagon, regular, has all the diagonals equal, so the probability of selecting the shortest= probability of selecting the longest=1 (they are all the same).

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Re: What is the probability of randomly selecting one of...... [#permalink]

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15 Apr 2013, 23:08
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 1838 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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Re: What is the probability of randomly selecting one of...... [#permalink]

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15 Apr 2013, 23:40
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...

its just 40%, youre overthinking it..

in an octagon, there are 8 points. to see how many diagonals there are for each point, just take the 7 other points and subtract 2 for the points directly adjacent. this leaves 5.

since its regular, the closest two diagonals will be the same length... aka the shortest. 2 out of 5. also, since it's regular this ratio applies to every point.

2/5 = 40%

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Re: What is the probability of randomly selecting one of...... [#permalink]

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16 Apr 2013, 00:51
Thank You Karishma. I got the point.
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Re: What is the probability of randomly selecting one of......   [#permalink] 16 Apr 2013, 00:51
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