Area alone is not enough information. Imagine, for instance, a parallelogram with a shorter side of  and a longer side of . The diagonal would be well above 

However, with the perimeter, the smaller and larger sides must add up to one half of it, .

The longer diagonal reaches its maximum with the larger internal angle widens towards  degrees, and the parallelogram flattens into a line. Using the law of cosines, this translates to:

.

At its max, the diagonal could be no greater than 

To find the area we need the length and width of the rectangle. We can use II together with I to make an equation for perimeter with only one unknown.

So we need both to solve.

Solve for  and then go back to find  and then with that you can find the area of the base and you are finished.

When asked to find the width of a rectangle we will need to use both statemests together.

For Statement I) we can use the perimeter formula.

Now, for Statement II) we will use the length of the diagonal along with the Pythagorean Theorem.

From here you can solve the perimeter equation in terms of either l or w. Then you can use substitution into the Pythagorean Theorem to solve for a possible width. 

Statement 1: 

Statement 1 is sufficient to answer the question

Statement 2: 

Statement 2 is also sufficient to answer the question

Statement 1:      

Recall the formula to find the perimeter of a rectangle. Substitute in the given information and solve.

Statement 2: 

Recall the formula for the area of a rectangle. Substitute in the given information and solve.

Each statement alone is sufficient to answer the question.

The length of one diagonal alone does not prove the parallelogram to be a rectangle, nor do the lengths of the sides.

Suppose we know all of these lengths, though. Since  is a parallelogram, if , then . 

The sides  and diagonal   form a triangle  with sidelengths 25, 60, and 65. The parallelogram is a rectangle if and only if  is a right angle; therefore, we must determine whether the conditions of the Pythagorean Theorem hold:

This is true;  is a right angle and  is a rectangle.

Therefore, both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.

In order to find the diagonal, we must know the sides of the rectangle or know whether the triangles ADC or ABD have special angles.

Statement 1 alone doesn't let us calculate the hypothenuse of the triangles, because we only know one side.

Statement 2 alone is sufficient because it allows us to find all angles of the triangles inside of the rectangle. We can see that they are special triangles with angles 30-60-90. Any triangle with these angles will have its sides in ratio , where  is a constant. Here, , knowing this, we can calculate the length of the hypothenuse, also the diagonal, which will be .

Hence, statement 2 is sufficient.

I) Gives us the length of ASOF's diagonal. This by itself does not give us any way of finding the other sides.

II) Gives us one side length. From there we can use the perimeter to find the other side length and then the area.

Therefore, Statement II is sufficient to answer the question.

The length of a diagonal of a rectangle can be calculated like a hypotenuse using the Pythagorean Theorem provided we have information about the lengths of the rectangle. 

Statement 1 tells us that both triangles ADC and ABD have their angles in ratio , which means that their sides will have length in ratio , where  is a constant. We can't tell however what length the diagonal will be. 

Statment 2 tells us that side AC is 1. From there we can't conclude anything. Indeed, rectangle ABCD might as well be a square or a very thin rectangle, we don't know.

Both statements together however, allow us to tell that  and therefore that the diagonal will be 2.

Hence, both statements together are sufficient.

To solve this, we need information about the lengths of the sides or to whether triangles ADB and ACD are special triangles.

Statement 1 tells us that CDA has length 3. This is not enough and we still don't know whether the rectangle is of a special type of rectangle.

Statement 2 tells us that triangles ADB and ACD are special triangles, indeed, they have their angles in ratio . That means that their sides will be in ratio . Now we don't need to know what is constant  , since it will cancel out in the ratio.

Therefore, statement 2 alone is sufficient.