Eight points are equally spaced on a circle

Eight points are equally spaced on a circle. If 4 of the 8 points are to be chosen at random, what is the probability that a quadrilateral having the 4 points chosen as vertices will be a square? 1/70, 1/35, 1,7, 1/4, 1/2

3 Explanations

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Shubhangi Kukreti

Is there any other way of finding out how many squares can be made without drawing them out?

Aug 23, 2018 • Comment

David Recine

Absolutely. If you're "fluent" enough in the rules of geometry, you can recognize that each distinct square will have its own 4 points, and that none of the four points on one square will act as a point ont he other square. 4 unique points in square 1 + 4 unique points in square 2 = the 8 vertices. So there can be only 2 squares.

From there, you can complete the problem as seen in Chris's explanation video.

With that said, 8 vertices in a circle = 2 inscribed squares is a pretty obscure geometry rule. So drawing out the circle is the way that most students would be able to find or check the rule to make sure they're confident before doing the further calculations.

Aug 25, 2018 • Reply

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Eric Kuzmenko

You can also do this: 1×(3÷7)×(2÷6)×(1÷5)

Explanation:
probability of choosing the first vertex is 1
TIMES
probability of choosing one of corresponding vertices of the square is then 3/7
TIMES
probability of choosing another (3rd) vertex is 2/6
TIMES
probability of choosing the last (4th) vertex is 1/5

You don't multiply this by two (although there are two different squares you can make) because the first vertex you've picked as immediately limited you to one of the squares.

Aug 12, 2014 • Comment

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Chris Lele