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.

From statement (1), we know that the side of S is 4, and then we can calculate the diagonal of S using the Pythagorean theorem: . The diagonal of S is the diameter of C. Therefore, we can calculate the circumference by using . From statement (2), we know that the side of S is 6, and then we can calculate the diagonal of S using the Pythagorean theorem:

The diagonal of S is the diameter of C. Therefore, we can calculate the circumference by using .

We are given the area and diameter of a circle and asked to find the circumference. We know that diameter is twice the length of a radius, so we also have our radius.

Given the following equations:

We can see that knowing either diameter or area will allow us to find the circumference.

Thus: Each statement alone is enough to solve the question.

All we need to find circumference is the radius. 

I) Gives us the radius squared, so we could find circumference with I.

II) Tells us the slope of the tangent line at a given point is undefined. Only vertical lines have undefined slope. The tangent line is perpendicular to the radius, so we can find our radius by drawing a picture and comparing the location of the center to the location of the tangent line.

So either statement will be sufficient.

We are asked to find the circumference  of Circle  and are given the diameter and the area. We also know that .  Taking each statement individually:

1.) The radius  is  and we know that . Therefore, Statement 1 is sufficient to solve for the circumference of the circle by itself.

2.) The area  of the outside circle is , but we cannot use this to determine the circumference of Circle  because we don't know where  is inside the larger outside circle.  

We are asked to find the circumference  of Circle  and are given the diameter and the area. We also know that .  Taking each statement individually:

1.) The diameter  is  and we know that the radius , so . Therefore, Statement 1 is sufficient to solve for the circumference of the circle by itself.

2.) The area  of Circle  is , so we can determine that the radius . Since the circumference , Statement 2 is is sufficient to solve for the circumference of the circle by itself.