Is the radius of the circle greater than 3?

Yes, your understanding is correct.

Maximum distance between two points on a circle is the length of its diameter (so when these points are the endpoints of a diameter), so the max distance = diameter = 2 radii.

So for example if we are told that the distance between the two points on a circle is 10cm then we can be sure that the diameter is more than or equal to 10 (or the radius is more than or equal to 10/2=5).

Next, the formula to calculate the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\).

Now, back to the original question:

Is the radius of the circle greater than 3?
1) (2,4) and (5,10) lie on the circle --> the distance between these points is \(d=\sqrt{(2-5)^2+(4-10)^2}=\sqrt{45}>6\), so the diameter of this circle must be more than 6, thus the radius must be more than 3 --> answer to the question is YES. Sufficient.

2) (2,4) and (4,1) lie on the circle --> the distance between these points is \(d=\sqrt{(2-4)^2+(4-1)^2}=\sqrt{13}<6\), so the diameter of this circle may or may not be more than 6. Not sufficient.

Answer: A.