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Math Expert V
Joined: 02 Sep 2009
Posts: 60646
Is the point Q on the circle with center C ?  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 57% (02:03) correct 43% (02:05) wrong based on 609 sessions

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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.

DS30602.01
OG2020 NEW QUESTION

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Math Expert V
Joined: 02 Aug 2009
Posts: 8336
Re: Is the point Q on the circle with center C ?  [#permalink]

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1
Bunuel wrote:
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.

DS30602.01
OG2020 NEW QUESTION

A great question from OG..

Refer to the sketch attached..

(1) R is a point on the circle and the distance from Q to R is equal to the distance from Q to C.
QR=QC. This means Q is EQUIDISTANT from C and R
Since C is center and R is on the circle, the line L bisecting CR will depict the possible values of Q.
Where the line L intersects circle, Q is on the circle, otherwise NOT on the circle..
Insuff

(2) S is a point on the circle and the distance from Q to S is equal to the distance from S to C.
QS=SC=radius. This means S is EQUIDISTANT from C and Q, and SQ is equal to the radius of the circle
Since C is center and S is on the circle, the circle B with center at S will depict the possible values of Q.
Where the two circles intersect, Q is on the circle, otherwise NOT on the circle..
Insuff

Combined..
Now, Q will be at the intersection of line L and circle B.
two cases

Case I..If R and S are different point..
Q can be in the center and not on the circle.
Case II..If R and S are same point..
Q will be on the circle.
Insuff

E
Attachments circles.png [ 10.99 KiB | Viewed 5501 times ]

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EMPOWERgmat Instructor V
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Re: Is the point Q on the circle with center C ?  [#permalink]

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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

GMAT assassins aren't born, they're made,
Rich
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Re: Is the point Q on the circle with center C ?  [#permalink]

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1
Bunuel wrote:
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.

DS30602.01
OG2020 NEW QUESTION

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

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Re: Is the point Q on the circle with center C ?  [#permalink]

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Bunuel wrote:
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.

DS30602.01
OG2020 NEW QUESTION

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?
Math Expert V
Joined: 02 Aug 2009
Posts: 8336
Re: Is the point Q on the circle with center C ?  [#permalink]

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vanam52923 wrote:
Bunuel wrote:
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.

DS30602.01
OG2020 NEW QUESTION

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
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Re: Is the point Q on the circle with center C ?  [#permalink]

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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
_________________ Re: Is the point Q on the circle with center C ?   [#permalink] 13 Oct 2019, 20:23
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