In order to find the diagonal of a rectangle we need the length of both sides.

We can't find the lengths with just the area or just the perimeter, but by using them both together we can make a small system of equations with two unknowns and two equations.

Then we can solve for each side and use the Pythagorean Theorem to find our diagonal.

When a square and rectangle have the same perimeter, the square will have a larger area because having 4 equal sides maximizes the area. However, from statement 1, it is impossible to tell if the rectangle is also a square. When the information from statement 2 is combined, we can conclude that the rectangle is not also a square.

To find the rectangle, you need the length and the width.

If you know the diagonal  and the angle  it forms with one of the longer sides, you can use trigonometry to find both length and width:

From there, the area follows.

If you know only the diagonal, you have insufficient information; the length and width can vary according to that angle. If you only know the angle, you can discern the proportions of the sides, but not the actual lengths.

The portion of the rectangle to the right of the -axis has area ; to the left, . From Statement 1 alone, since , , and the portion of the rectangle on the right is greater than the portion on the left. However, this is all we can determine.

By a similar argument, from Statement 2 alone, the portion of the rectangle above is greater than the portion below, but this is all we can determine.

From both statements together, however, we can compare the portions of the rectangles in the four quadrants. The areas of each are:

Quadrant I: 

Quadrant II: 

Quadrant III: 

Quadrant IV: 

Since  and ,  is the greatest of the four quantities, and we see that Quadrant I includes the lion's share of the rectangle.

The figure referenced is below:

Assume Statement 1 alone.  is a diagonal of the rectangle, and also a diameter of the circle. Statement 1 gives the area of the circle, from which the radius, and, subsequently, , can be calculated. However, infinitely many rectangles of different areas can be constructed in this circle, so without any further information, it is not clear what the sidelengths are - and what the area is.

Assume Statement 2 alone. This statement only gives the length of one side. Without any further information, the area of the rectangle is unknown.

Now assume both statements are true.  can be calculated, and  is given, so the Pythagorean Theorem can be used to find . The area of the rectangle is the product , so the two statements together are sufficient.

The figure referenced is below (note that the figure itself assumes Statement 2, but this is not known from Statement 1):

Assume Statement 1 alone. A diagonal of a rectangle inscribed inside a circle is a diameter of the circle. Statement 1 gives the area of the circle, from which the radius, and, subsequently, , can be calculated. However, infinitely many rectangles of different areas can be constructed in a given circle, so without any further information, it is not clear what the sidelengths are - and what the area is.

Assume Statement 2 alone. It follows that all of the sides of the rectangle/square are congruent, but without the common sidelength, the area of the square cannot be calculated.

Assume both statements.  can be calculated, and the area of the square can be calculated to be .

Assume both statements. Then from Statement 1, it follows that:

There are two pairs of odd primes that add up to 18 - (5,13), in which case the area is 65, and (7,11), in which case the area is 77. The two statements together are inconclusive.

A a rectangle with sides of length 1 and 4 and a rectangle with sides of length 2 and 3 both have perimeter 10, but they have different areas ( 4 and 6, respectively), making Statement 1 alone inconclusive. Statement 2 is inconclusive, there being infinitely many primes.

Assume both statements.

Then 

Since  and  are both prime integers, one must be 2 and the other must be 3 (1 and 4 cannot be a possibility, since 1 is not a prime). It does not matter which is which, so the numbers can be multiplied to obtain area 6.

To know the perimeter of a rectangle, we need to know length and width. 

Statement (1) provides Area, which does not provide the necessary information. NOT SUFFICIENT

Statement(2) provides one side, but we still need the other to determine perimeter. NOT SUFFICIENT.

Both together are sufficent however since if we know one side and the area, we can find the remaining side using division, and then use the two side lengths to find the perimeter. 

Let  be the sidelength of the square. Then its perimeter is .

From Statement 1 alone, the length of the rectangle is , and, if  is the width, the perimeter of the rectangle is . Therefore, we can prove that the rectangle has the greater perimeter.

From Statement 2 alone, the width of the rectangle is , and its perimeter is at least  - but unless we know the length, we do not know whether the total perimeter is greater than, equal to, or less than .