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A circle in a coordinate plane has a center at point A and a diameter

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A circle in a coordinate plane has a center at point A and a diameter  [#permalink]

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New post 17 Jun 2015, 05:54
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A circle in a coordinate plane has a center at point A and a diameter of 6. If points B and C also lie in the same coordinate plane, is point B inside the circle?

(1) The distance between point A and point C equals 2.
(2) The distance between point B and point C equals 2.

Kudos for a correct solution.

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Re: A circle in a coordinate plane has a center at point A and a diameter  [#permalink]

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New post 17 Jun 2015, 07:30
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Bunuel wrote:
A circle in a coordinate plane has a center at point A and a diameter of 6. If points B and C also lie in the same coordinate plane, is point B inside the circle?

(1) The distance between point A and point C equals 2.
(2) The distance between point B and point C equals 2.

Kudos for a correct solution.


Question : Is point B inside the circle?

Diameter = 6

CONCEPT: Point B will be inside the circle if The distance AB is less than Radius

Statement 1: The distance between point A and point C equals 2.

i.e. AC= 2

i.e. Point C is in the Circle but the relation of Point C with Point B is still unknown

Hence, NOT SUFFICIENT

Statement 2: The distance between point B and point C equals 2.

But the relation of Point C or Point B with Point A is still unknown

Hence, NOT SUFFICIENT

Combining the two statement

AC = 2 and BC =2 i.e. Distance AB can vary from 0 to 4

Hence, NOT SUFFICIENT

Answer: Option
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Re: A circle in a coordinate plane has a center at point A and a diameter  [#permalink]

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New post 17 Jun 2015, 07:38
2
Bunuel wrote:
A circle in a coordinate plane has a center at point A and a diameter of 6. If points B and C also lie in the same coordinate plane, is point B inside the circle?

(1) The distance between point A and point C equals 2.
(2) The distance between point B and point C equals 2.

Kudos for a correct solution.


We have a circle of radius 3 here..
1) statement one tells us the distance of C from A and that A is in circle but nothing about B.. insuff
2) statement one tells us the distance of C from B but nothing about the circle.. insuff

Combined it gives us distance between A and C and also between B and C..
but we can still not say anything about relation of A and B as B could lie on any side of C , basically on any circumference of a circle with center C and radius 2..
insuff
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Re: A circle in a coordinate plane has a center at point A and a diameter  [#permalink]

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New post 17 Jun 2015, 09:24
1
Bunuel wrote:
A circle in a coordinate plane has a center at point A and a diameter of 6. If points B and C also lie in the same coordinate plane, is point B inside the circle?

(1) The distance between point A and point C equals 2.
(2) The distance between point B and point C equals 2.

Kudos for a correct solution.


My attempt :

Radius of circle is 3. A is center of circle. So any point which is at distance less than 3 will be inside circle. So basically to answer the question " is point B inside the circle?" We must know what is distance between A (center of circle) and point B.

Statement 1 :
It gives distance between A and C, not distance between A and B. Hence not sufficient.

Statement 2 :
It gives distance between B and C, not distance between A and B. Hence not sufficient.


Statement 1 and 2 combine.
d(A,C) = 2
d(B,C) = 2

point B can be anywhere in the Co-ordinate plane and hence we will not know if B is inside circle.

Say for example.
B is same line, and C is between A and B, then d(A,B) = 4 and B will lie outside of circle

B is in same line and A and B coincides. B lie inside the circle.


Hence Option E
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Re: A circle in a coordinate plane has a center at point A and a diameter  [#permalink]

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New post 17 Jun 2015, 11:30
1
using statement 1 alone , we can see C is inside circle . but no info about B ------ insufficient

Using statement 2 alone , no info about any of B and C location with respect to centre A ------ insufficient

Combining both ,
points can be on radius .. as A-C-B in this case B goes outside circle
or Point A and C on radious and B can be anywhere inside circle at distance of 2 from C.

So even now we can not find B's position uniquely.

So insufficient . Answer E.
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Re: A circle in a coordinate plane has a center at point A and a diameter  [#permalink]

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New post 22 Jun 2015, 05:10
Bunuel wrote:
A circle in a coordinate plane has a center at point A and a diameter of 6. If points B and C also lie in the same coordinate plane, is point B inside the circle?

(1) The distance between point A and point C equals 2.
(2) The distance between point B and point C equals 2.

Kudos for a correct solution.


MANHATTAN GMAT OFFICIAL SOLUTION:

Is B inside the circle?
Image
Using Rubber Band Geometry concepts, we know from the problem that the exact locations of points A, B, and C do not matter—only the relative locations of the points matter. Therefore, we can arbitrarily assign point A to a specific location (in this case, we should choose the origin of the coordinate plane), and draw a circle with a radius of 3 units around it.

(1) The distance between A and C is 2.
Image
Statement (1) does not tell us anything about point B, so it is not sufficient. However, it does tell us that A and C are 2 units apart, so Statement (1) enables us to place point C anywhere along the grey circle.

(1) The distance between B and C is 2.
Image
Statement (2) does not tell us anything about point B relative to point A, so it is not sufficient. However, it does constrain point B to be exactly 2 units away from wherever point C is. Thus we can imagine point C at the center of a circle of size 2, with point B somewhere on the circle around it.

(1) and (2) combined
Image
Finally, we can combine these two statements to see that depending upon where we draw point C, and then draw point B relative to point C, point B may be inside the dotted circle, and it may not be.

The correct answer is E.

Notice that we never proved the insufficiency of Statements (1) and (2) combined using algebra or any computation—but then again we didn't need to! Rather, we deduced it by drawing the information that was given to us. We reasoned visually that under the constraints, we could draw a scenario in which point B was inside the circle. We could also draw another scenario in which point B was outside of the circle. Done! We achieved a visual proof of the answer.

That's what Rubber Band Geometry is all about: testing scenarios for Geometry problems without the need to plug in numbers or use algebra. All we need is a visual environment that can be manipulated—one that preserves all key constraints and freedoms in the problem, and allows us to see them and test them. Rubber Band Geometry provides that environment for us.

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Re: A circle in a coordinate plane has a center at point A and a diameter  [#permalink]

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New post 22 Mar 2016, 17:43
Bunuel wrote:
A circle in a coordinate plane has a center at point A and a diameter of 6. If points B and C also lie in the same coordinate plane, is point B inside the circle?

(1) The distance between point A and point C equals 2.
(2) The distance between point B and point C equals 2.

Kudos for a correct solution.


1. nothing about B
2. nothing about the relationship between these two points and the origin (point A).

1+2.
B can be inside or B can be outside.

E
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Re: A circle in a coordinate plane has a center at point A and a diameter  [#permalink]

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New post 10 Dec 2019, 23:46
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Re: A circle in a coordinate plane has a center at point A and a diameter   [#permalink] 10 Dec 2019, 23:46
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