If Points A and B are Randomly Placed on the Circumference GMAT Problem Solving

Question: 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. \(\frac{1}{4}\)
2. \(\frac{1}{3}\)
3. \(\frac{1}{2}\)
4. \(\frac{2}{3}\)
5. \(\frac{5}{6}\)

Answer:

Approach Solution (1):

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

Correct Option: D

Approach Solution (2):

Let O be the center of the circle. If we fix point A somewhere on the circumference of the circle, and if B is anywhere on the right or left of point A such that angle AOB is no greater than 60 degrees, then the length of chord AB will be no greater than 2. That is, since B can be to the right or left of point A, then B can be anywhere on a 60 + 60 = 120-degree arc (where A is the center of this arc) such that chord AB is no longer than 2. However, if B is anywhere outside this 120-degree arc (i.e., B is anywhere on the 240-degree arc that is outside of the 120-degree arc), then chord AB will be longer than 2. Since the circle has 360 degrees, the probability B is on the 240-degree arc is 240/360 = 2/3.
Correct Option: D

Approach Solution (3):

Let us assume that the centre is A and the two ends of the chord are B and C
Let us first assume that the length of the chord is 2

If the length of the chord has to be 2 to start with. The triangle created by drawing lines from the two ends of the chord to the centre would be an equilateral triangle
Which means angle BAC would be 60 degrees. If the angle BAC is less than 60 then the length of the chord would be less than 2 and if it is more than 60 it would be greater than 2

This means that there are 120 possibilities for angle BAC where the length of BC would be greater than 2
The probability would therefore be 120/180 = 2/3

Correct Option: D

“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?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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