Is the point Q on the circle with center C ?

Hi All,

We're asked if the point Q is on the circle with center C. This is a YES/NO question and can be approached with some logic (and a few examples/drawings might help).

(1) R is a point on the circle and the distance from Q to R is equal to the distance from Q to C.

Since C is the CENTER of the circle, and R is on the circumference, we know that CR is a RADIUS of the circle. With the information in Fact 1, point Q can be ANY point that is equidistance from R and C.
IF....
-we have an equilateral triangle, with R and Q on the circumference, then the answer to the question is YES.
-we have point Q in the exact middle of the radius CR, then the answer to the question is NO.
Fact 1 is INSUFFICIENT

(2) S is a point on the circle and the distance from Q to S is equal to the distance from S to C.

Since C is the CENTER of the circle, and S is on the circumference, we know that CS is a RADIUS of the circle. With the information in Fact 2, we know that the length of QS is the SAME length as the RADIUS.
IF....
-we have an equilateral triangle, with S and Q on the circumference, then the answer to the question is YES.
-we have point Q in any other position that is exactly one radius in length from point S, then the answer to the question is NO.
Fact 2 is INSUFFICIENT

Combined, we know...
R is a point on the circle and the distance from Q to R is equal to the distance from Q to C.
S is a point on the circle and the distance from Q to S is equal to the distance from S to C.

With both Facts, we can create two different examples:
-we have two equilateral triangles, with Q on the circumference and R and S on opposite sides of Q, then the answer to the question is YES.
-we have an isosceles triangle, with Q OUTSIDE of the circle and CR as the non-equal side, then the answer to the question is NO.
Combined, INSUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich

1) We know that point R is on the circle. We can infer that point Q is on the perpendicular bisector of segment CR. If we draw a figure, then we can see that this perpendicular bisector intersects the circle at two points, but it also has an infinite number of points not on the circle. Thus, we can't get a definite answer to the original question. \(\implies\) Insufficient

2) We know that point S is on the circle. It's possible that point S is on the perpendicular bisector of segment CQ, as in the case in which point Q is on the circle. However, it's also possible that C and Q are not distinct points. Thus, we can't get a definite answer to the original question. \(\implies\) Insufficient

1&2) It's possible that point Q is on the circle at either intersection point of the circle and the perpendicular bisector of segment CR. However, it's also possible that point Q is inside the circle at the midpoint of segment CR. Thus, we can't get a definite answer to the original question. \(\implies\) Insufficient

Answer: E

Bunuel
chetan2u
EMPOWERgmatRichC

can we consider the case when point is outside circle?
we are considering either on circle or iside circle but point can be ouside circle too .is it correct?

Yes, the point could be anywhere - outside, on or inside-, but we have to just check whether the point Q is ON the circle

Hi vanam52923,

As with any DS question, we have to think carefully about what we KNOW vs. what we DON'T know. For example, when a DS question tells us that X is an integer, that does NOT necessarily mean that it's a positive integer (X could be 0 or a negative).

In this question, the prompt tells us NOTHING about Point Q, so it might be on the circle... or it might be "inside" the circle or "outside" of the circle. You have to consider those possibilities to correctly answer this question.

GMAT assassins aren't born, they're made,
Rich

Quick question : When the question stem says 'Point ON the circle' does it mean just on the circumference or does it mean that point can be on the circumference or it can be inside the circle as well ?

On the circle = ON the circumference.
IN the circle = INSIDE the circle.

Just a quick point to add.
As in SC question we do the vertical scanning, here also doing so we can see that we need not to do the same thing for Statement 2 as it is roughly the same statement except the highlighted part i.e. Q can be on the imaginary circle with center S with all the three conditions possible. For Statement 1 Q can take either of the positions - outside the circle OR on the circle OR inside the circle - on the line perpendicular to RC.

So, we just need to check for combination.

Combining St. 1 and St. 2
Q can be a common point for both statements when R and S are same point or it may take different position - a point where line perpendicular to RC and circle with center S intersect - when R and S are different point.

Answer E.

Solved this one. But gave me a doubt, If OG Says there are two points Q and C does that mean they cannot be positioned at the same coordinates?. I haven't seen anyone using a case where Q coincides with C to cancel option C and pick E. I remember a similar GMATPrep Question using modulus and numberline which used the fact they could actually be coinciding.

I tried solving the question without any diagrams. My rationale was that since Q could be a point that could lie outside/inside/on the circle and since none of the options invalidated these possibilities either individually or together, therefore E must be the correct answer.

Answer: Option E

Video solution by GMATinsight

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

Here is my approach to the problem. Hopefully the visual helps understand better:

Video solution from Quant Reasoning:
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