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Center of the circle

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Center of the circle

by gmattesttaker2 » Sun Dec 01, 2013 12:51 pm
Hello,

Can you please assist with this:

Point P is inside circle X. If A, B, and C are three different points on the
circumference of circle X, is point P the center of circle X?

(1) PA = PB
(2) PA = PC

OA: E

I was thinking that it should be D since for PA to be equal to PB or PC each has to be the radius. Can you please assist?

Thanks,
Sri
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Source: — Data Sufficiency |

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by [email protected] » Sun Dec 01, 2013 7:41 pm
Hi Sri,

This DS question will likely be made easier IF you draw some pictures. I'm going to offer some hints and then let you try to solve this problem again.

The "obvious" possibility is that P is the center of the circle. Try drawing that picture with P, A and B (with A and B on the circumference of the circle). Next, pick a point for P OTHER than the center of the circle; could you draw an A and B in such a way that PA and PB would be the same length? Now try drawing a picture with P, A, B and C. These drawings will provide you with the proof that the correct answer is as listed.

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by Matt@VeritasPrep » Sun Dec 01, 2013 11:31 pm
I think the OA is wrong - a point inside a circle that is equidistant from three points on the circumference (i.e. if PA = PB = PC, as we have here) should be the center.

I'll link you to a nice proof at https://www.artsofliberty.org/sites/defa ... er%203.pdf and take a screenshot of part of the proof:

Image

Image

An easy (GMATy) way of thinking about it, however, is to consider what you have when you take the two statements together. If PA = PB = PC, then you have SOME circle centered at P that goes to A, B, and C, with PA, PB, and PC each the radius of that circle. But A, B, and C are ALSO supposedly on the circumference of some "other" circle that isn't centered at P ... and how can that be? (Try visualizing it and it should hopefully make sense.)
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