GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 27 Jun 2019, 02:13

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

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

Author Message
TAGS:

### Hide Tags

CEO
Joined: 12 Sep 2015
Posts: 3787
If two points, A and B, are randomly placed on the circumference  [#permalink]

### Show Tags

Updated on: 26 May 2017, 13:50
8
Top Contributor
23
00:00

Difficulty:

95% (hard)

Question Stats:

42% (02:14) correct 58% (02:19) wrong based on 192 sessions

### HideShow timer Statistics

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

*kudos for all correct solutions

_________________
Test confidently with gmatprepnow.com

Originally posted by GMATPrepNow on 16 Feb 2017, 11:59.
Last edited by GMATPrepNow on 26 May 2017, 13:50, edited 1 time in total.
CEO
Joined: 12 Sep 2015
Posts: 3787
Re: If two points, A and B, are randomly placed on the circumference  [#permalink]

### Show Tags

16 Feb 2017, 16:20
6
Top Contributor
2
GMATPrepNow 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

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

_________________
Test confidently with gmatprepnow.com
Current Student
Joined: 28 Jan 2017
Posts: 31
Location: Chile
Concentration: General Management, Strategy
GMAT 1: 710 Q50 V35
GPA: 3.2
Re: If two points, A and B, are randomly placed on the circumference  [#permalink]

### Show Tags

Updated on: 17 Feb 2017, 05:05
7
2
(Another method)

Fix point A in the circumference (let's choose the north pole)

From A, draw the two possible chords of length 6 ('left' and 'right')
Ending points are X and Y.
Join A, X and Y to the center O.

We have two equilateral triangles (each side is 6), so angle XOY is 120 degrees.

Now, point B of our 'chord grater than 6' can be chose anywhere outside arch YX. As inside is 120, outside is 240 degrees.

So the probability is 240 (degrees)/ 360 (degrees) = 2/3 letter E

Originally posted by pitroncoso on 16 Feb 2017, 12:44.
Last edited by pitroncoso on 17 Feb 2017, 05:05, edited 1 time in total.
##### General Discussion
Intern
Joined: 03 Jan 2017
Posts: 30
Re: If two points, A and B, are randomly placed on the circumference  [#permalink]

### Show Tags

Updated on: 16 Feb 2017, 12:52
Tough call...

Tried and failed to do...need to brush up my skills

Originally posted by Jayeshvekariya on 16 Feb 2017, 12:35.
Last edited by Jayeshvekariya on 16 Feb 2017, 12:52, edited 2 times in total.
Intern
Joined: 17 Dec 2014
Posts: 23
Location: India
Schools: IMD Jan'18 (D)
GMAT 1: 710 Q48 V38
WE: Engineering (Energy and Utilities)
Re: If two points, A and B, are randomly placed on the circumference  [#permalink]

### Show Tags

17 Feb 2017, 11:47
1
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

Intern
Joined: 17 Nov 2016
Posts: 24
Re: If two points, A and B, are randomly placed on the circumference  [#permalink]

### Show Tags

25 May 2017, 05:49
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?
CEO
Joined: 12 Sep 2015
Posts: 3787
Re: If two points, A and B, are randomly placed on the circumference  [#permalink]

### Show Tags

25 May 2017, 06:19
Top Contributor
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?

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.

Does that help?

Cheers,
Brent
_________________
Test confidently with gmatprepnow.com
Intern
Joined: 17 Nov 2016
Posts: 24
Re: If two points, A and B, are randomly placed on the circumference  [#permalink]

### Show Tags

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.

Does that help?

Common sense. Thanks a lot
Senior Manager
Joined: 29 Jun 2017
Posts: 449
GPA: 4
WE: Engineering (Transportation)
Re: If two points, A and B, are randomly placed on the circumference  [#permalink]

### Show Tags

10 Sep 2017, 07:11
1
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.

therefore P(total) = 120+120 / 360 = 240/360 = 2/3
Attachments

IMG_3717.JPG [ 1.63 MiB | Viewed 2575 times ]

_________________
Give Kudos for correct answer and/or if you like the solution.
Senior Manager
Joined: 29 Jun 2017
Posts: 449
GPA: 4
WE: Engineering (Transportation)
Re: If two points, A and B, are randomly placed on the circumference  [#permalink]

### Show Tags

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.
_________________
Give Kudos for correct answer and/or if you like the solution.
Non-Human User
Joined: 09 Sep 2013
Posts: 11449
Re: If two points, A and B, are randomly placed on the circumference  [#permalink]

### Show Tags

18 Sep 2018, 03:18
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: If two points, A and B, are randomly placed on the circumference   [#permalink] 18 Sep 2018, 03:18
Display posts from previous: Sort by