Opposite angles of a parallelogram are congruent, and if the parallelogram is inscribed, both angles are inscribed as well. Congruent inscribed angles on the same circle intercept congruent arcs; since the two congruent arcs together comprise a circle, each intercepted arc is a semicricle. This makes the angles right angles, and this forces a parallelogram inscribed in a circle to be a rectangle.

Statement 1 alone tells us that this is also a square, and that its sides have length 20. The diagonal of a square, which is also a diameter of the circle that circumscribes it, has length  times that of a side, or ; half this, or , is the radius of the circle.

Statement 2 alone gives a diagonal of the rectangle, which, again, is enough to determine the radius of the circle.

The figure referenced is below:

By symmetry,  is one third of a circle. Therefore, its length is one third of the circumference, so, if Statement 1 alone is assumed, the circumference can be determined to be ; this can be divided by  to yield radius .

Statement 2 yields no helpful information; from the body of the problem,  can already be deduced to be two thirds of a circle, or, equivalently, an arc of measure

.

From Statement 2 alone, , a diagonal of the square, measures . The diameter of the circle is equal to the length of a diagonal of an inscribed square, so the radius of the circle is equal to half this, or .

From Statement 1 alone, since the area of the square is 100, its sidelength is the square root of this, or 10. By the 45-45-90 Theorem, a diagonal of the square measures  times this, or , which makes Statement 2 a consequence of Statement 1. Therefore, it follows again that the circle has radius .

If a right triangle is inscribed inside a circle, the hypotenuse of the triangle is a diameter of the circle; therefore, its length is the diameter, and half this is the radius. 

Statement 1 alone does not give the hypotenuse of the triangle, or, for that matter, any of the sidelengths. Statement 2 alone gives one sidelength, but does not state whether it is the hypotenuse or not.

Assume both statements are true. Since  in right triangle , then either  and , or vice versa. In either event, , being opposite the  angle, is the short leg of a 30-60-90 triangle, and, by the 30-60-90 Theorem, the hypotenuse is twice its length. This is twice 18, or 36. This is the diameter of the circle, and the radius is half this, or 18.

Knowing neither the center alone, as given in Statement 1, nor two points alone, as given in Statement 2, is sufficient to find the radius of the circle. 

Assume both statements to be true. Knowing the center from Statement 1 and one point on the circle, as given in Statement 2, is enough to determine the radius - use the distance formula to find the distance between the two points and, equivalently, the radius.

Statement 2 is unhelpfful as it gives no information about the circles - only a triangle which is not given in that statement to have any connection to the circles. Statement 1 alone is unhelpful in that it only identifies two sides of a triangle as  diameters of the circles without giving their lengths.

Now assume both statements to be true. The hypotenuse of a right triangle, which Statement 2 gives as , must be the longest side, so  has greater length than . This means that the circle with  as a diameter, Circle 1, must have a greater diameter than one with  as a diameter, Circle 2. Since Circle 1 has the greater diameter, it has the greater radius.

Statement 1 alone yields insufficient information, since, as seen in the diagram below, the circles that circumscribe a square and a triangle with the same sidelength have different sizes:

Statement 2 is insufficient since it gives no hints about the size of the hexagon.

Assume both statements. The six radii of a regular hexagon divide it into six equilateral triangles, by symmetry; therefore, the radius of a regular hexagon is equal to its sidelength, which is given in Statement 1 as 10. Since the radius of a regular hexagon is equal to that of the circle in which it is inscribed, the circle has radius 10.

This can be seen by examining the figure below:

Let  be the measure of  and  be radius.

From Statement 1 alone, the area of the shaded sector is 

However, we have no other information, so we cannot determine the value of the radius.

From Statement 1 alone, the length of the arc of the shaded sector  is

Again, we have no other information, so we cannot determine the value of the radius.

Assume both statements hold. From Statements 1 and 2, we have, respectively,

and

If we divide, we get the radius:

.

The actual form of the equation of a circle is 

,

where  is the location of the center, and  is the radius. 

The radius of the circle in the equation

is therefore . We need to know the value of  in the equation.

Assume both statements are true. Then we can add the equations to get :

But without further information, we cannot determine .

If a right triangle is inscribed inside a circle, the hypotenuse of the triangle is a diameter of the circle; therefore, its length is the diameter, and half this is the radius. However, each statement alone gives us the length of only one side, and does not clue us in to which side is the hypotenuse.

Assume both statements are true, however. The hypotenuse must always be the longest side of the right triangle, so if two sides have the same length, they must be the legs. A right triangle with congruent legs is a 45-45-90 triangle, and by the 45-45-90 Theorem, its hypotenuse must measure  the length of a leg. Since each leg of  measures 100, the length of the hypotenuse, and the diameter of the circle, are , and the radius is half this, or .