Problem Solving

If 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 will be at least 6 inches?

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

Source: GMAT Prep Now
Difficulty level: 700+

Answer: E

Brent@GMATPrepNow wrote:If 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 will be at least 6 inches?

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

regor60's solution is perfect.
Below is a very similar solution (with a few diagrams to help students visualize the solution)

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.




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.




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




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.



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)



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



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


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: E

Hi Matt,

I checked the Mosteller link, and it seems to discuss the issues related to randomly selecting a chord, whereas my question involves choosing 2 random points that will create a chord.

The Mosteller link (in part c) seems to suggest that, if we create a chord by choosing 2 random points (as my question suggests), then the answer to my original question is [spoiler]2/3[/spoiler]

Am I missing something?

Cheers,
Brebt