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If points A and B are randomly placed on the circumference of a circle

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If points A and B are randomly placed on the circumference of a circle  [#permalink]

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New post 27 Oct 2010, 17:47
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If points A and B are randomly placed on the circumference of a circle with radius 2, what is the probability that the length of chord AB is greater than 2?

1/4
1/3
1/2
2/3
3/4

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Re: If points A and B are randomly placed on the circumference of a circle  [#permalink]

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New post 27 Oct 2010, 18:37
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Look at the figure. Let A be the first point which can be anywhere... Now AB and AC are chords of length 2 cm each giving you equilateral triangles BOA and COA. (Center of the circle is the point O)
If your second point is anywhere on arc BAC, the length of the chord will be less than or equal to 2. Else, it will be greater than 2.
Since the arc BAC subtends 120 degrees at the center, probability of chord length less than 2 cm is 1/3 and greater than 2 cm is 2/3
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Re: If points A and B are randomly placed on the circumference of a circle  [#permalink]

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New post 27 Mar 2018, 10:57
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Hi All,

First off, a "chord" is defined as a line that connects any two points on the circumference of a circle. The longest chord on any circle is the diameter of the circle, but any two points can form a chord.

To solve this question, try this….For this question, we have a radius of 2. Pick ANY point on the circumference. If you draw a cord with a length of 2 from that point, you can then draw two radii to those two points to form a triangle. What type of triangle has three sides that are all the same length? An equilateral triangle, which has angles of 60/60/60.

Now, from your starting point, draw another cord of length 2 in the OTHER direction. You'll end up repeating the steps above and you'll end up with another 60/60/60 triangle.

Those two central angles: 60 + 60 = 120 degrees. From your original starting point - to every point OUTSIDE of that 120 degrees - will create a cord that is GREATER than 2. So, 240 degrees of the circle will give you the result that you're looking for. 240/360 = 2/3

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Re: If points A and B are randomly placed on the circumference of a circle  [#permalink]

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New post 07 Dec 2018, 08:30
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shrive555 wrote:
If points A and B are randomly placed on the circumference of a circle with radius 2, what is the probability that the length of chord AB is greater than 2?

1/4
1/3
1/2
2/3
3/4


We'll begin by arbitrarily placing point A somewhere on the circumference.
Image



So, we want to know the probability that a randomly-placed point B will yield a chord AB that is at least 2 cm long.
So, let's first find a location for point B that creates a chord that is EXACTLY 2 cm long.
Image



There's also ANOTHER location for point B that creates another chord that is EXACTLY 2 cm long.
Image



IMPORTANT: For chord AB to be greater than or equal to 2 cm, point B must be placed somewhere along the red portion of the circle's circumference.
Image


So, the question really boils down to, "What is the probability that point B is randomly placed somewhere on the red line?"
To determine this probability, notice that the 2 cm chords are the same length as the circle's radius (2 cm)
Image


Since these 2 triangles have sides of equal length, they are equilateral triangles, which means each interior angle is 60 degrees.
Image


The 2 central angles (from the equilateral triangles) add to 120 degrees.
This means the remaining central angle must be 240 degrees.
Image

This tells us that the red portion of the circle represents 240/360 of the entire circle.
So, P(point B is randomly placed somewhere on the red line) = 240/360 = 2/3

Answer: D

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Brent
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Re: If points A and B are randomly placed on the circumference of a circle   [#permalink] 07 Dec 2018, 08:30
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