To prove two figures similar, we must prove that their corresponding angles are congruent and that their corresponding sides are in proportion.

Assume both statements are true. We show that they provide insufficient information by examining two scenarios.

Case 1: Trapezoid  Trapezoid , then  and , so the conditions of both statements are met; also, since the trapezoids are congruent, they are also similar.

Now examine isosceles trapezoid  below, in which , , , and  are positioned on the bases so that  and .

Since  and , Quadrilateral  is a parallelogram, and ; similarly, . Therefore, Trapezoid  is also isosceles. Also, by the Corresponding Angles Theorem,  and , and the conditions of both statements are met. However, corresponding sides are not in proportion, since , but ; consequently, the trapezoids are not similar.

Similar rectangles (or any shape for that matter) are the same shape but can be different sizes. What that means is that their side lengths all follow a common ratio; if one pair of corresponding sides follow the ratio 2:1, then all corresponding sides must follow the same ratio.

Statement I gives us the perimeter and one side of .

Statement II gives us the area and one side of .

We can find all the sides of both rectangles, but we need both statements to do so. Once we have all side lengths, we can compare them to see if they follow the same ratios.

IF  has perimeter of 16 and one side is 3, we can find the other side using the following:

If  has area of 44 and side of 6, the other side can be found via the following:

Compare the ratios of the sides to find out whether the two rectangles are similar:

Therefore, the rectangles are not similar.

Statement 1:  The area is 1.

Given the area of a square is sufficient to find the diagonal of the square.  The formula to find the side length of the square is:

The Pythagorean theorem can then be used to solve for the length of the diagonal.

Statement 2:  The square is inscribed inside of a circle with all 4 corners touching the edge of the circle, and the circle has an area of 1.

This statement is also sufficient to solve for the diagonal.  With all 4 corners of the square touching the circle, the diagonal of the square is simply the length from one of the square's opposing 90 degree angles to its opposite angle, which is the diameter of the circle.  

Given the area of the circle, use the formula  to determine the diameter of the circle, which is also the diagonal of the square.

Therefore:

Find the length of the diagonal of rectangle DEFT

I) Side DE is 3 times the length of side TD

II) The perimeter of the rectangle is 

We need to find the diagonal of a rectangle. If we work back from there, we can see that to find the diagonal, we will need to the lengths of both sides of the rectangle.

I) Gives us the relationship between the two sets of sides:

II) Tells us the perimeter, for which we can write and equation:

Now, use the clue from I) to simplify our equation in II) to one variable:

So now that we have our side lengths, use Pythagorean Theorem to find the diagonal:

With a ratio of sides of 2:3, statement 1 tells us PS and QR are 9 units long. Knowing the ratio we can determine both PQ and SR are 6. Since rectangles have four 90 degree angles, we can use the Pythagorean Theorem to solve for the length of SQ: . Therefore, statement 1 alone is sufficient.

With a ratio of sides of 2:3, statement 2 tells us PQ and SR are 6 units long. Knowing the ratio we can determine both PS and QR are 9 units long. Since rectangles have four 90 degree angles, we can use the Pythagorean Theorem to solve for the length of SQ:  . Therefore, statement 2 alone is sufficient.

Therefore, the correct answer is  EACH statement ALONE is sufficient.

By definition, a parallelogram is a quadrilateral with 2 sets of parallel sides and diagonals that bisect each other.

From statement 1 we cannot determine the length of AC, since ABCD is only defined as a parallelogram and not necessarily a rectangle (which is a special parallelogram with equal diagonals). Therefore, statement 1 alone is not sufficient.

From statement 2, we can determine that  since the definition of a parallelogram states the diagonals bisect each other. Therefore, Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

With the information from statement 1, we don't know the measurements of any parts of PQRS. Therefore,  Statement 1 alone is not sufficient.

With the information from statement 2, we know that  , but since we have no confirmed definition of the shape of PQRS, we can't extrapolate the length of SQ to any other length.  Therefore, statement 2 alone is not sufficient.

Even if we look at both statements 1 and 2 together, quadrilateral PQRS looks like a rectangle, but we are only told it is a parallelogram. The figures on the GMAT are not drawn to scale, therefore your eyes cannot be trusted. No difinitive length of PR can be determined. Therefore, statements 1 and 2 together are not sufficient.

Therefore, the correct answer is Statements (1) and (2) TOGETHER are NOT sufficient.

To find the diagonal of a rectangle, we need the length of two sides.

I) Gives us the length of one side.

II) Lets us find the length of the next side.

Use I) and II) with Pythagorean Theorem to find the diagonal. 

Conversely, recognize that we are making a 3/4/5 Pythagorean Triple and see that the last side is 30 fathoms.

Statement 1:  The perimeter of the square is known.

A square has four equal sides.  Write the perimeter formula for squares.

The side length of a square is a fourth of the perimeter.

Statement 2:  A secondary square with a known area encloses the primary square where all 4 corners of the primary square touch each of secondary square's edges.

If the primary square corners touch each of the secondary square edges, they must touch at the midpoint of each edge.  Since Statement 2 mentions that the secondary square area is known, it is possible to solve for the edge length and the diagonal of the secondary square.  Write the formula for the area of a square.

The diagonal of the secondary square can be solved by using the Pythagorean Theorem.

The side length of the secondary square also must equal the diagonal of the primary square.

Therefore:

Each of the statements, 1 and 2, provide something integral to calculating the length of a diagonal: an angle and the length of the sides connected to its vertex. The diagonal would be the third leg of the resulting triangle, and can be calculated using the law of cosines:

Since this is a parellelogram, knowing one angle allows us to know all the angles.