Apparently incorrect answer in the Geometry Flash Cards Q-8

[LazyNK]

Hi,
I was going through the practice flash cards on geometry. I think that the answer to the data sufficiency question is wrong.

I have attached the snapshot of the question here.



<PSR=30, The question is whether the statement <POR=60 sufficient to say that "O" is the center of the circle.

The answer given is that it is sufficient.

However, in my opinion, it is not sufficient, because the relation I know is as follows :
The angle created by an arc at the circumference of the circle is half the angle created by that arc at the center.
BUT
If the angle created by an arc at the circumference of the circle is half the angle created by that arc at any other point inside the circle , it doesn't imply that that point is the center of the circle.

Infact, one can clearly see that it is possible to draw 60 degree angles at many other ooints inside the circle in the given problem - All of them would not be the centers of the circle.

Please confirm.
Regards
NK

[jnelson0612]

Hi,
I was going through the practice flash cards on geometry. I think that the answer to the data sufficiency question is wrong.

I have attached the snapshot of the question here.



<PSR=30, The question is whether the statement <POR=60 sufficient to say that "O" is the center of the circle.

The answer given is that it is sufficient.

However, in my opinion, it is not sufficient, because the relation I know is as follows :
The angle created by an arc at the circumference of the circle is half the angle created by that arc at the center.
BUT
If the angle created by an arc at the circumference of the circle is half the angle created by that arc at any other point inside the circle , it doesn't imply that that point is the center of the circle.

Infact, one can clearly see that it is possible to draw 60 degree angles at many other ooints inside the circle in the given problem - All of them would not be the centers of the circle.

Please confirm.
Regards
NK

NK,
The explanation on the card is correct. It is a geometric rule that if I have:
1) An arc created by a point touching the exact opposite side
2) That same arc which is created by an angle opening from a point at the center
3) Then the center angle is exactly twice the measure of the angle at the far side.

I think where you are running into trouble is envisioning other angles around this circle, if I am understanding your question. What is relevant here is that both of these angles open to the exact same arc. Because the center one is exactly twice the far one, that center one must be fixed at the center of the circle. Hope this helps!

[LazyNK]

Hi Jamie,
Thanks for your reply, but I'm sorry - I still don't agree. What I said was :
The angle created by an arc at the circumference of the circle is half the angle created by that arc at the center.
BUT
If the angle created by an arc at the circumference of the circle is half the angle created by that arc at any other point inside the circle , it doesn't imply that that point is the center of the circle.

In short, if A implies B, then it can't be said surely that B implies A (unless the relationship is symmetric). I agree with you when you say that when O is the center of the given circle, <POR=60. But given question says that <POR=60. Can it imply that O is the center of the circle ? In my opinion, NO.
You could probably visualize, that given the problem, you can find multiple points X inside the given circle at which <PXR=60.

Another way to look at this is as follows :
1.) Draw a circle through the points P, O and R ( the center of this circle will be the centroid of the triangle POR).
2.) Line PR will project the same angle as <POR, on all points on this new circle lying on same side of PR as O. Many of these points will be inside the given circle.
3.) So, there can be more than one point inside the given circle at which PR projects an angle=60.
4.) Since all those points can't be center of the given circle, it is not sufficient to say that if <POR=60 , O is the center of the circle.
Hence the given statement is not sufficient.
Can you please relook ?
Regards
NK

[tim]

Nishu, you are absolutely correct. The card is in error; you're right that the converse of the rule does not hold. I'll alert MGMAT to the error so it can be corrected when someone gets around to it..