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# In a plane, points P and Q are 20 inches apart. If point R is randomly

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Math Expert
Joined: 02 Sep 2009
Posts: 60630
In a plane, points P and Q are 20 inches apart. If point R is randomly  [#permalink]

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20 Nov 2019, 00:39
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Difficulty:

95% (hard)

Question Stats:

32% (01:54) correct 68% (01:48) wrong based on 62 sessions

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In a plane, points P and Q are 20 inches apart. If point R is randomly chosen from all the points in the plane that are 20 inches from P, what is the probability that R is closer to P than it is to Q?

A. $$0$$

B. $$\frac{1}{4}$$

C. $$\frac{1}{3}$$

D. $$\frac{1}{2}$$

E. $$\frac{2}{3}$$

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VP
Joined: 19 Oct 2018
Posts: 1294
Location: India
Re: In a plane, points P and Q are 20 inches apart. If point R is randomly  [#permalink]

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20 Nov 2019, 03:46
6
2
Locus of R is a circle, whose center is P and whose radius is 20 inches.

Distance of R from P is 20 inches

R is closer to P if R lies anywhere on the highlighted(Red) portion of the circle

Probability= 240/360=2/3

Bunuel wrote:
In a plane, points P and Q are 20 inches apart. If point R is randomly chosen from all the points in the plane that are 20 inches from P, what is the probability that R is closer to P than it is to Q?

A. $$0$$

B. $$\frac{1}{4}$$

C. $$\frac{1}{3}$$

D. $$\frac{1}{2}$$

E. $$\frac{2}{3}$$

Are You Up For the Challenge: 700 Level Questions

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##### General Discussion
Intern
Joined: 28 Sep 2019
Posts: 5
Re: In a plane, points P and Q are 20 inches apart. If point R is randomly  [#permalink]

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21 Nov 2019, 05:01
IMO OA is 0
All the points of R form a circle with P as the center of the circle. Point Q and any point R always lie on the circle. so the Point P is always at equal distance from R & Q and there is no chance where R can be closer to P than Q. So the combinations are 0 hence the probability is 0
Senior Manager
Joined: 16 Feb 2015
Posts: 259
Location: United States
Concentration: Finance, Operations
Re: In a plane, points P and Q are 20 inches apart. If point R is randomly  [#permalink]

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21 Nov 2019, 11:25
saik1993 wrote:
IMO OA is 0
All the points of R form a circle with P as the center of the circle. Point Q and any point R always lie on the circle. so the Point P is always at equal distance from R & Q and there is no chance where R can be closer to P than Q. So the combinations are 0 hence the probability is 0

saik1993, Your all explanation is wrong.

Please see the above post, explained very good by nick1816

Hope you will read his answer, if you didn't get it. we will explain you.

Intern
Joined: 28 Sep 2019
Posts: 5
Re: In a plane, points P and Q are 20 inches apart. If point R is randomly  [#permalink]

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22 Nov 2019, 03:11
rajatchopra1994
I read the solution from nick1816

I have a few questions,

1. If point R is on the circle, with P as the center, Isn't the distance between them is now the radius of the circle =20
and Q is also 20 inches away from P?

2. From the diagram, Let us consider the other two points be A & B. I can see that PA= 20 & PB= 20 as they are radius,
but how do we get the other side (QA & QB length as 20? why did we assume the angle BPA as 120 and line segment PQ bisects the angle BPA? Why can't the angles be 180 or 60 only?

Thank you.
Intern
Joined: 28 Sep 2019
Posts: 5
Re: In a plane, points P and Q are 20 inches apart. If point R is randomly  [#permalink]

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25 Nov 2019, 02:17
If I choose a point on the circle which is not highlighted in red, it is 20 inches away from P just as in the red highlighted area?
VP
Joined: 19 Oct 2018
Posts: 1294
Location: India
Re: In a plane, points P and Q are 20 inches apart. If point R is randomly  [#permalink]

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25 Nov 2019, 04:51
In the non-highlighted potion, R is closer to Q (less than radius of circle) than it is to P. We are only looking for the portion where R is closer to P than it is to Q.

saik1993 wrote:
If I choose a point on the circle which is not highlighted in red, it is 20 inches away from P just as in the red highlighted area?
Intern
Joined: 28 Sep 2019
Posts: 5
Re: In a plane, points P and Q are 20 inches apart. If point R is randomly  [#permalink]

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25 Nov 2019, 05:04
Thank you all, read the question wrong.
Re: In a plane, points P and Q are 20 inches apart. If point R is randomly   [#permalink] 25 Nov 2019, 05:04
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