AbeNeedsAnswers wrote: ↑Tue Apr 30, 2019 5:45 pm
Is the point Q on the circle with center C ?
(1) R is a point on the circle and the distance from Q to R is equal to the distance from Q to C.
(2) S is a point on the circle and the distance from Q to S is equal to the distance from S to C.
E
Source: Official Guide 2020
Solution:
Question Stem Analysis:
We need to determine whether point Q is on the circle C, i.e., whether point Q is on the circumference of circle C.
Statement One Alone:
From statement one, we see that Q is on the perpendicular bisector of radius RC. Since the perpendicular bisector of a radius of a circle intersects the circle at two points. Q might or might not be on circle C (that is, if Q is one of the two intersection points, then it’s on the circle; otherwise, it is not). Statement one alone is not sufficient.
Statement Two Alone:
From statement two, we see that Q is on a circle with the same radius as circle C but centered at S. Since such a circle intersects circle C at two points, Q might or might not be on circle C (that is, if Q is one of the two intersection points, then it’s on the circle; otherwise, it is not). Statement two alone is not sufficient.
Statements One and Two Together:
The point Q may or may not be on the circle even when we consider both statements together.
To come up with a scenario where point Q is on the circle, choose points R and S such that the angle RCS is 120 degrees. Choose point Q to be the point where the bisector of the angle RCS meets the circle. Notice that RQC and QCS are both equilateral triangles, and RC and SC are both radii of the circle. Thus, RQ = QC = QS = SC.
For the scenario where point Q is not on the circle, recall that point Q must be on the perpendicular bisector of radius RC. Let S be the one of the points this perpendicular bisector meets the circle. Draw a circle of radius SC = RC with center at S. This circle will intersect the perpendicular bisector in two points and neither of these points will be on the original circle. Notice that: 1) QR is equal to QC because Q is on the perpendicular bisector of RC, and 2) QS is equal to SC because Q is on the circle with center S and radius SC.
As we can see, even when we assume both statements, the point Q may or may not be on the circle with center C.
Answer: E
