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Is the point Q on the circle with center C ?

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

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New post 26 Apr 2019, 02:48
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A
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C
D
E

Difficulty:

  95% (hard)

Question Stats:

46% (02:05) correct 54% (02:10) wrong based on 94 sessions

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

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New post 26 Apr 2019, 23:50
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
circles.png [ 10.99 KiB | Viewed 581 times ]


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

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New post 16 May 2019, 14:14
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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

Final Answer:

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Re: Is the point Q on the circle with center C ?   [#permalink] 16 May 2019, 14:14
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