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If points A and B are randomly placed on the circumference o

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If points A and B are randomly placed on the circumference o [#permalink] New post 01 Dec 2018, 11:01
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If points A and B are randomly placed on the circumference of a circle with a radius of 2 cm, what is the probability that the length of chord AB is greater than 2 cm?

A. \(\frac{1}{4}\)

B. \(\frac{1}{3}\)

C. \(\frac{1}{2}\)

D. \(\frac{2}{3}\)

E. \(\frac{3}{4}\)
[Reveal] Spoiler: OA

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Re: If points A and B are randomly placed on the circumference o [#permalink] New post 06 Dec 2018, 03:59
In trigonometry, we use Law of Cosines to find out the angle of a triangle when 3 sides are known. Here, two sides of triangle are 2,2 and third side should be less than 2. If, in a triangle, two equal sides (lines) are at 120 degrees, the sum of two sides (or the third side) will be same as the other two side. So in our question the angle should be above 120 degree. So probability of 120 degree in a 360 degree circle is
120/360 =1/3 where the length of chord will be less than 2.
But for the chord being greater than 2, the probability is
1 - 1/3
2/3
It's D
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Re: If points A and B are randomly placed on the circumference o [#permalink] New post 07 Dec 2018, 07:30
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Carcass wrote:
If points A and B are randomly placed on the circumference of a circle with a radius of 2 cm, what is the probability that the length of chord AB is greater than 2 cm?

A. \(\frac{1}{4}\)

B. \(\frac{1}{3}\)

C. \(\frac{1}{2}\)

D. \(\frac{2}{3}\)

E. \(\frac{3}{4}\)


We'll begin by arbitrarily placing point A somewhere on the circumference.
Image



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.
Image



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



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.
Image


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)
Image


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


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

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: D
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Re: If points A and B are randomly placed on the circumference o   [#permalink] 07 Dec 2018, 07:30
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