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P, Q, and R are three points in a plane, and R does not lie

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P, Q, and R are three points in a plane, and R does not lie [#permalink] New post 24 Jan 2016, 15:19
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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
[Reveal] Spoiler: OA

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Re: P, Q, and R are three points in a plane, and R does not lie [#permalink] New post 24 Jan 2016, 15:21
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Solution

First 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 \(B\)
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Re: P, Q, and R are three points in a plane, and R does not lie [#permalink] New post 19 Jun 2017, 14:44
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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)
Image

One obvious point that is equidistant from P and Q is this point...
Image


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
Image

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.
Image

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.
Image
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
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Re: P, Q, and R are three points in a plane, and R does not lie [#permalink] New post 09 May 2018, 15:06
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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
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Re: P, Q, and R are three points in a plane, and R does not lie [#permalink] New post 02 Nov 2018, 10:41
Intricate! Can we expect to see such questions on GRE?
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Re: P, Q, and R are three points in a plane, and R does not lie [#permalink] New post 02 Nov 2018, 11:20
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yes. For sure.

it is an official question from the Official Guide.

Regards
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Re: P, Q, and R are three points in a plane, and R does not lie [#permalink] New post 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)
Image

One obvious point that is equidistant from P and Q is this point...
Image


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
Image

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.
Image

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