Source: Official Guide Revised GRE 2nd Ed. Part 9; Section 5; #23

10

# P, Q, and R are three points

P, Q, and Rare 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? It contains no points., It contains one point., It contains two points., It is a line., It is a circle.

### 3 Explanations

1

Anna Roberts

What happens if R did lie on the line PQ?
Then how many set of points will be same distance from all the points?

Jan 6, 2018 • Comment

David Recine, Magoosh Tutor

In that case, there would be 0 points that are the same distance form all three. As mentioned in the answer explanation on page 543 of the Revised GRE OG 2nd edition, is *two* points lie on the same line, any equidistant points equidistant to those two points would have to lie on a line that bisects the two points perpendicularly. In fact, all points on the bisecting line would be equidistant from the two points.

As a result, if R, P, and Q were on the same line, there could be an infinite number of equidistant points for any *two* of R, P, and Q, but not for all three. This is because you can only make one bisecting perpendicular line that is halfway between two points on the The perpendicular bisecting lines between RP, RQ, and QP would all be perpendicular to the same line. So they would all be parallel and never intersect. With on intersection point for these lines, no equidistant points would ever meet for R, Q, and P.

9

Suresh Kumar

It's an equilateral triangle PQR with a dot in the middle which is equidistant from P,Q and R. Hence B.

Aug 12, 2014 • Comment

11

Chris Lele

Dec 8, 2012 • Comment