Eight points are equally spaced on a circle. If 4 of the 8 points are

Question

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?

A. 1/70
B. 1/35
C. 1/7
D. 1/4
E. 1/2

imeanup · Answer

ext{Probabilty of selecting 4 point out of 8 points} = \dfrac{1}{^8C_4} = \dfrac{1}{70}

ext{No of ways to select 4 points in 8 points} = \dfrac{8}{4} = 2

ext{Total Probability} = \dfrac{2}{70} = \dfrac{1}{35}

B

Vaishvii · Answer

How is itt 1/35? There is only one way to draw a square, right?

Sajjad1994 · Answer

OE

For questions involving geometry, it is often helpful to draw a figure representing the information in the question as accurately as possible. The figure below shows a circle with 8 equally spaced points, labeled A through H, and quadrilateral , which is one of the many quadrilaterals that have 4 of the 8 equally spaced points as vertices.

The probability that a quadrilateral having the 4 points chosen as vertices will be a square is equal to the following fraction.

The number of squares that can be drawn using 4 of the 8 points as vertices/The number of quadrilaterals that can be drawn using 4 of the 8 points as vertices

To calculate the desired probability, you need to determine the number of squares and the number of quadrilaterals that can be drawn using 4 of the 8 points as vertices.

To determine the number of quadrilaterals, first note that since the 8 points lie on a circle, every subset of 4 of the 8 points determines a unique quadrilateral. Therefore, the number of quadrilaterals that can be drawn using 4 of the 8 points as vertices is equal to the number of ways of choosing 4 points from the 8 points shown. The number of ways of choosing 4 points from the 8 points shown (also called the number of combinations of 8 objects taken 4 at a time) is equal to \(\frac{8!}{4!(8-4)!}\). You can calculate the value of this expression as follows.

\(\frac{8!}{4!(8-4)!}= \frac{8.7.6.5.4!}{4.3.2.1.4!}=70\)

Thus, there are 70 quadrilaterals that can be drawn using 4 of the 8 points as vertices.

Because the points are equally spaced around the circle, there are only 2 squares that can be drawn using 4 of the 8 points as vertices, namely ACEG and BDFH, as shown in the following figures.

Therefore, the probability that the quadrilateral will be a square is \(\frac{2}{70}\), or \(\frac{1}{35}\), and the correct answer is Choice B.

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Eight points are equally spaced on a circle. If 4 of the 8 points are

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