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?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.

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.

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

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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