The diameter of a circle with an inscribed rectangle is equal to the length of the diagonal of the rectangle; once this diameter is found, it can be divided by 2 to yield the radius.

Statement 1 alone gives this length, from which the radius can be found to be . Statement 2 alone gives only the length of one set of opposite sides, from which the length of the diagonal cannot be determined.

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.

Assume Statement 1 alone. The area of a right triangle is half the product of the lengths of the legs, which is 36. However, this is not enough to determine the length of the hypotenuse. 

For example, if the legs measure 8 and 9, the triangle has area

By the Pythagorean Theorem, the hypotenuse measures

,

which is the diameter of the circle; half this, or , is the radius of the circle.

If the legs measure 6 and 12, the triangle has area

By the Pythagorean Theorem, the hypotenuse measures

,

which is the diameter of the circle; half this, or , is the radius of the circle.

Therefore, Statement 1 is inisufficient to give the radius.

Assume Statement 2 alone. The hyppotenuse of a right triangle must be longer than either leg, so it is impossible for either of the two equally long sides  and  to be the hypotenuse of ; they must be its legs. Since the legs are of the same length,  is a 45-45-90 triangle, and by the 45-45-90 Theorem, hypotenuse  has length  that  of a leg, or . This is the diameter of the circle, and the radius is half this, or 

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 , making Statement 2 alone sufficient to answer the question - and Statement 1 unhelpful.

Assume both statements are true. Since we have no way to determine which, if either, of the legs of  is the longer, we have no way to compare the diameters, and, consequently, no way to compare the radii, of the circles with those segments as diameters.

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, from the two statements, it cannot be determined which segment is the hypotenuse. 

If   is the hypotenuse, then the radius is half its length; since , the radius is 10.

If  is the hypotenuse, then, since the hypotenuse is the longest side of a right triangle,  - that is, . The radius is greater than 10.

Therefore, the radius depends on which side is the hypotenuse; since that is not clear, the radius cannot be determined.

The diameter of a circle with an inscribed rectangle is equal to the length of a diagonal of the rectangle , which, given the length  and width , can be found using the Pythagorean Theorem:

Statement 1 alone gives insufficient information. For example, a 20 by 10 rectangle has area , and a 40 by 5 rectangle has area . 

The first rectangle has diagonals of length

.

The second rectangle has diagonals of length

,

Since the diagonals of the rectangles differ, so do the diameters, and, consequently, the radii, of the circles.

Statement 2 alone gives insufficient information for a similar reason. For example, a 20 by 10 rectangle has perimeter , and a 25 by 5 rectangle has perimeter . Again, the first rectangle has  diagonals of length . The second has diagonals of length

.

Now, assume both statements to be true. We are looking for two numbers whose product is 200 and whose sum is 30 (since the perimeter is twice the sum, or 60). The only such pair of numbers can be found by trial and error to be 20 and 10, so these are the length and width of the rectangle. As shown before, a rectangle with these dimensions has diagonals of length . This is the diameter of the circle in which it is inscribed, so half this, or , is the radius.

Statement 1 alone only gives one point through which the circle passes, so no information can be determined about the other points or about the size or location of the circle. A similar argument holds for the insufficiency of Statement 2.

Now assume both statements are true. The circle has exactly three intercepts, but it is given that there are two -intercepts -  and one other point - and two -intercepts -  and one other point. The unidentified -intercept and the unidentified -intercept must be one and the same, and the only possible way this can happen is for this common point to be the origin . Since three points define a circle, we can now identify the unique circle through the points , , and , and we can figure out its radius.

The two statements together only give information about angle measures; arc degree measures can be deduced from this information but not any arc lengths or side lengths. Without this information, we cannot obtain the radius of this circle.