Aug 23 08:00 PM PDT  10:00 PM PDT Get Magoosh's App for Free Questions, Free Video lessons, and more. Aug 25 09:00 PM PDT  10:00 PM PDT Target Test Prep is giving 10 lucky winners 4 months of FREE access to our toprated GRE Quant course. Our giveaway contest is open for 10 days only, so enter today for your chance to levelup your prep! Aug 26 08:00 PM PDT  09:00 PM PDT We have a growing team of trained online experts, but much of our online GRE tutoring is delivered by a true GRE expert, MyGuru's Director of Online Tutoring, Stefan. Aug 26 08:00 PM PDT  11:00 PM PDT Learn how to evaluate your profile, skills, and experiences to determine if, when, and where you should apply to graduate school. Aug 28 08:00 PM PDT  11:00 PM PDT Free Video Modules Within each of the 16 learning modules (Geometry, Statistics, Sentence Equivalence, etc.) that comprise the course, you'll find plenty of free videos to help you make an informed purchase. Aug 30 08:00 PM PDT  09:00 PM PDT 7+ POINT GRE SCORE IMPROVEMENT GUARANTEE Sep 01 08:00 PM PDT  09:30 PM PDT Whether you want to take a free practice test, attend a deepdive session about the test itself, or experience the power of live instruction, we’ve got you covered.
Author 
Message 
TAGS:


Founder
Joined: 18 Apr 2015
Posts: 7744
Followers: 143
Kudos [?]:
1593
[0], given: 7034

P, Q, and R are three points in a plane, and R does not lie [#permalink]
24 Jan 2016, 15:19
Question Stats:
96% (01:52) correct
3% (00:00) wrong based on 27 sessions
P, Q, and R are three points in a plane, and R does not lie on line PQ. Which of the following is true about the set of all points in the plane that are the same distance from all three points? A) It contains no points. B) It contains one point. C) It contains two points. D) It is a line. E) It is a circle. Practice Questions Question: 23 Page: 463 Difficulty: hard
_________________
Get the 2 FREE GREPrepclub Tests




Founder
Joined: 18 Apr 2015
Posts: 7744
Followers: 143
Kudos [?]:
1593
[0], given: 7034

Re: P, Q, and R are three points in a plane, and R does not lie [#permalink]
24 Jan 2016, 15:21
SolutionFirst consider just two of the three points, say P and Q, and the set of points in the plane that are the same distance from them. Clearly the midpoint of line segment PQ is such a point. Are there others? You may recall from geometry that the points on the line that bisects PQ and is perpendicular to PQ are all the points that are equidistant from P and Q. Similarly, the points in the plane that lie on the perpendicular bisector of line segment PR are all the points that are equidistant from points P and R. Because R does not lie on line PQ, line segments PQ and PR do not lie on the same line, and so their respective perpendicular bisectors are not parallel. Therefore, you can conclude that the two perpendicular bisectors intersect at a point. The point of intersection is on both perpendicular bisectors, so it is equidistant from P and Q as well as from P and R. Therefore, the point of intersection is equidistant from all three points. Are there any other points that are equidistant from P, Q, and R ? If there were, they would be on both perpendicular bisectors, but in fact only one point lies on both lines. The answer is \(B\)
_________________
Get the 2 FREE GREPrepclub Tests



GRE Instructor
Joined: 10 Apr 2015
Posts: 2303
Followers: 72
Kudos [?]:
2190
[6]
, given: 26

Re: P, Q, and R are three points in a plane, and R does not lie [#permalink]
19 Jun 2017, 14:44
6
This post received KUDOS
Carcass wrote: P, Q, and R are three points in a plane, and R does not lie on line PQ. Which of the following is true about the set of all points in the plane that are the same distance from all three points? A) It contains no points. B) It contains one point. C) It contains two points. D) It is a line. E) It is a circle. Practice Questions Question: 23 Page: 463 Difficulty: hard Tricky question!!!I thought I'd create some images to go along with Carcass' solution. Let's first consider 2 points (P and Q) One obvious point that is equidistant from P and Q is this point... Also recognize that, if we draw two circles of equal radii with their centers at PQ, the 2 points of intersection will also be equidistant from P and Q In fact, all points on this red line are equidistant from P and Q. The red line (as Carcass mentioned above) is the perpendicular bisector of points P and Q. At this point, let's add a 3rd point (R) and draw the perpendicular bisector of points R and P in blue. All points on the blue line will be equidistant from R and P. So, the green point (where the blue line and red line intersect) must be equidistant from R and P, AND it must equidistant from P and Q. So, that green point must be equidistant from P, Q and R Since there is only 1 such green point, the correct answer is B. Cheers, Brent
_________________
Brent Hanneson – Creator of greenlighttestprep.com Sign up for my free GRE Question of the Day emails



Target Test Prep Representative
Status: Head GRE Instructor
Affiliations: Target Test Prep
Joined: 09 May 2016
Posts: 161
Location: United States
Followers: 6
Kudos [?]:
151
[2]
, given: 0

Re: P, Q, and R are three points in a plane, and R does not lie [#permalink]
09 May 2018, 15:06
2
This post received KUDOS
Carcass wrote: P, Q, and R are three points in a plane, and R does not lie on line PQ. Which of the following is true about the set of all points in the plane that are the same distance from all three points?
A) It contains no points. B) It contains one point. C) It contains two points. D) It is a line. E) It is a circle. If R is not on line PQ, then the three points form the vertices of a triangle. Let’s call it triangle PQR. Since there is only one point that is equidistant (the same distance) from the three vertices of the triangle (and the point is actually called the circumcenter), choice B is the correct answer. Answer: B
_________________
5star rated online GRE quant self study course See why Target Test Prep is the top rated GRE quant course on GRE Prep Club. Read Our Reviews



Intern
Joined: 27 Oct 2018
Posts: 49
Followers: 0
Kudos [?]:
15
[0], given: 27

Re: P, Q, and R are three points in a plane, and R does not lie [#permalink]
02 Nov 2018, 10:41
Intricate! Can we expect to see such questions on GRE?



Founder
Joined: 18 Apr 2015
Posts: 7744
Followers: 143
Kudos [?]:
1593
[0], given: 7034

Re: P, Q, and R are three points in a plane, and R does not lie [#permalink]
02 Nov 2018, 11:20
yes. For sure. it is an official question from the Official Guide. Regards
_________________
Get the 2 FREE GREPrepclub Tests



Director
Joined: 09 Nov 2018
Posts: 509
Followers: 0
Kudos [?]:
39
[0], given: 1

Re: P, Q, and R are three points in a plane, and R does not lie [#permalink]
16 Nov 2018, 20:38
GreenlightTestPrep wrote: Carcass wrote: P, Q, and R are three points in a plane, and R does not lie on line PQ. Which of the following is true about the set of all points in the plane that are the same distance from all three points? A) It contains no points. B) It contains one point. C) It contains two points. D) It is a line. E) It is a circle. Practice Questions Question: 23 Page: 463 Difficulty: hard Tricky question!!!I thought I'd create some images to go along with Carcass' solution. Let's first consider 2 points (P and Q) One obvious point that is equidistant from P and Q is this point... Also recognize that, if we draw two circles of equal radii with their centers at PQ, the 2 points of intersection will also be equidistant from P and Q In fact, all points on this red line are equidistant from P and Q. The red line (as Carcass mentioned above) is the perpendicular bisector of points P and Q. At this point, let's add a 3rd point (R) and draw the perpendicular bisector of points R and P in blue. All points on the blue line will be equidistant from R and P. So, the green point (where the blue line and red line intersect) must be equidistant from R and P, AND it must equidistant from P and Q. So, that green point must be equidistant from P, Q and R Since there is only 1 such green point, the correct answer is B. Cheers, Brent Why don't we consider opposite side of the line PQ. I mean we can put D Both side of PQ. In that case we have two points. Please explain.



Intern
Joined: 13 May 2019
Posts: 2
Followers: 0
Kudos [?]:
3
[2]
, given: 0

Re: P, Q, and R are three points in a plane, and R does not lie [#permalink]
09 Aug 2019, 00:11
2
This post received KUDOS
Another useful way to think about this problem is to realize that three points determine a circle. We can draw a circle such that points P, Q, and R all lie on the circumference, with the circle's center being the same distance (radius) from all three points. Since the points are not colinear only one such circle is possible.




Re: P, Q, and R are three points in a plane, and R does not lie
[#permalink]
09 Aug 2019, 00:11





