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

 It is currently 16 Oct 2019, 12:36

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

Author Message
TAGS:

### Hide Tags

Manager
Status: Do and Die!!
Joined: 15 Sep 2010
Posts: 244
If points A and B are randomly placed on the circumference of a circle  [#permalink]

### Show Tags

27 Oct 2010, 17:47
1
9
00:00

Difficulty:

75% (hard)

Question Stats:

45% (01:51) correct 55% (01:48) wrong based on 97 sessions

### HideShow timer Statistics

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

_________________
I'm the Dumbest of All !!
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9705
Location: Pune, India
Re: If points A and B are randomly placed on the circumference of a circle  [#permalink]

### Show Tags

27 Oct 2010, 18:37
4
2
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
Attachment:

circle.jpg [ 9.53 KiB | Viewed 2281 times ]

_________________
Karishma
Veritas Prep GMAT Instructor

##### General Discussion
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15262
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: If points A and B are randomly placed on the circumference of a circle  [#permalink]

### Show Tags

27 Mar 2018, 10:57
1
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

GMAT assassins aren't born, they're made,
Rich
_________________
Contact Rich at: Rich.C@empowergmat.com

The Course Used By GMAT Club Moderators To Earn 750+

souvik101990 Score: 760 Q50 V42 ★★★★★
ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 4006
Re: If points A and B are randomly placed on the circumference of a circle  [#permalink]

### Show Tags

07 Dec 2018, 08:30
1
Top Contributor
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.

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.

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

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.

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)

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

Cheers,
Brent
_________________
Test confidently with gmatprepnow.com
Re: If points A and B are randomly placed on the circumference of a circle   [#permalink] 07 Dec 2018, 08:30
Display posts from previous: Sort by