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If two points, A and B, are randomly placed on the circumfer

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If two points, A and B, are randomly placed on the circumfer [#permalink] New post 26 May 2017, 12:48
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If two points, A and B, are randomly placed on the circumference of a circle with circumference 12pi inches, what is the probability that the length of chord AB is at least 6 inches?

(A) 1/(2pi)
(B) 1/pi
(C) 1/3
(D) 2/pi
(E) 2/3
[Reveal] Spoiler: OA

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GRE Instructor
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Joined: 10 Apr 2015
Posts: 1245
Followers: 46

Kudos [?]: 1135 [3] , given: 7

Re: If two points, A and B, are randomly placed on the circumfer [#permalink] New post 29 May 2017, 16:42
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GreenlightTestPrep wrote:
If two points, A and B, are randomly placed on the circumference of a circle with circumference 12pi inches, what is the probability that the length of chord AB is at least 6 inches?

(A) 1/(2pi)
(B) 1/pi
(C) 1/3
(D) 2/pi
(E) 2/3


Let's first determine the details of this circle.
For any circle, circumference = (diameter)(pi)

The circumference of the given circle is 12pi inches, so we can write: 12pi inches= (diameter)(pi)
This tells us that the diameter of the circle = 12 inches
It also tells us that the radius of the circle = 6 inches


Okay, now let's solve the question.
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 6 inches long.
So, let's first find a location for point B that creates a chord that is EXACTLY 6 inches.
Image



There's also another location for point B that creates another chord that is EXACTLY 6 inches.
Image



IMPORTANT: For chord AB to be greater than or equal to 6 inches, 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 6-inch chords are the same length as the circle's radius (6 inches)
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:
[Reveal] Spoiler:
E

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Re: If two points, A and B, are randomly placed on the circumfer   [#permalink] 29 May 2017, 16:42
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If two points, A and B, are randomly placed on the circumfer

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