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.
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!