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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?

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

pitroncoso'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

Re: If two points, A and B, are randomly placed on the circumference [#permalink]

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17 Feb 2017, 11:47

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Probability questions involving circles are best solved using area or angle subtended on the centre.

As 12pi is the area, diameter of circle is is 12 in. Taking the limiting case of 6 in chord - A chord of 6 inch will subtend 60 degrees on the the centre. (Equilateral Triangle with the radius)

Any chord more than 6 inch will subtend more than 60 degrees on the centre.

So the Probability of the chord being in that sector is 240/360 = 2/3

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

Can you please explain why you considered two points for B not just one? is not the answer 300/360 instead?

You bet. Once point A is placed on the circumference, there are TWO chords of length 6 that can be connected to point A: one chord is to the left of point A, and the other chord is to the right of point A. If we don't consider both points then we will be including a portion of the circle where the chord AB can be less than 6.

Re: If two points, A and B, are randomly placed on the circumference [#permalink]

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25 May 2017, 06:24

Quote:

You bet. Once point A is placed on the circumference, there are TWO chords of length 6 that can be connected to point A: one chord is to the left of point A, and the other chord is to the right of point A. If we don't consider both points then we will be including a portion of the circle where the chord AB can be less than 6.

Re: If two points, A and B, are randomly placed on the circumference [#permalink]

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10 Sep 2017, 07:11

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Quite simple ques see the figure attached 2pi R = 12 pi => R=6 when AB is joined ( length = 6 minimum ) consider AB as 6

the angle made by chord at center of the circle will be 60 degrees

Now consider same chord on opposite side same angle will be 60

so the angle 60 will be the angle on which any 2 points taken beyond AB will be less than 6

There fore 2 sides were there => 1 - 120/360 = 2/3

THIS WAS 1 METHOD

2ND METHOD:

see the picture attached between 2 chords of 6 in length 120 degree is the angle on 1 side on which any point taken suppose A and on the other side if any point taken between that 120 degree on the circle will make >= 6 chords' length.

Re: If two points, A and B, are randomly placed on the circumference [#permalink]

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10 Sep 2017, 09:34

Zoser wrote:

Quote:

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

Can you please explain why you considered two points for B not just one? is not the answer 300/360 instead?

Hello Zoser you can consider 2 parallel chords as explained in my answer. And find the extremities of 6 then calculate the angle, i guess it is self explanatory.
_________________

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