Write the formula for finding the area of a square given the diagonal.

Rearrange the equation so that the diagonal term is isolated.

Substitute the known area and simplify.

The diagonal of a square is simply the hypotenuse of a right triangle whose other two sides are the length and width of the square. Because all sides of a square are equal in length, this means the length and width are both  ,  which gives us a right triangle with a base of    and a height of  ,  for which the hypotenuse is the diagonal of the square. Applying the Pythagorean Theorem to find the length of the diagonal, we have:

So the length of the diagonal for a square with a side length of    is  .  In general, we could check the length of the diagonal for any square with side length  ,  and we would see that the diagonal length is always  .

The diagonal of a rectangle can be thought of as the hypotenuse of a right triangle whose base and height are the length and width of the rectangle, respectively.  This means we can use the Pythagorean Theorem to calculate the length of the diagonal for a rectangle:

The area of a rhombus is half the product of the lengths of its diagonals, which here are  and . This means

Since both diagonals have whole numbers as their lengths, and , we are looking for  to be a whole number that can be divided into 144 to yield a quotient less than . Equivalently, 

The quotient is 

Since we can multiply both sides by  to yield the inequality

,

we know that

, 

so we can eliminate 12 and 16.

Also, since , 10 is not correct, as  would not be a whole number.

If , then . Both diagonals have lengths that are whole numbers, and  is the longer diagonal. 9 is the correct choice.

A rhombus has four sides of equal length. Since Rhombus  has a right angle , it follows that, the rhombus being a parallelogram, all four angles are right angles, and, by definition, Rhombus  is a square. The length of a diagonal of a square is  times the length of a side; since the rhombus has perimeter 80, each side measures one fourth of this, or 20, and diagonal  has length .

The area of a rhombus is half the product of the lengths of its diagonals, which here are  and . This means

Therefore, we need to test each of the choices to find the pair of diagonal lengths for which this holds.

:

Area: 

Area: 

Area: 

Area: 

 is the correct choice.

The sides of a rhombus are all congruent; since the perimeter of Rhombus  is 64, each side measures one fourth of this, or 16. 

The referenced rhombus, along with diagonal , is below:

Since consecutive angles of a rhombus, as with any other parallelogram, are supplementary,  and  have measure ;  bisects both into  angles, making equilangular and, as a consequence, equilateral. Therefore, .

The referenced rhombus, along with diagonals  and , is below.

The four sides of a rhombus have equal measure, so each side has measure one fourth of the perimeter of 48, which is 12.

Since consecutive angles of a rhombus, as with any other parallelogram, are suplementary,  and  have measure ; the diagonals bisect  and  into  and  angles, respectively, to form four 30-60-90 triangles.  is one of them; by the 30-60-90 Triangle Theorem, ,

and

.

Since the diagonals of a rhombus bisect each other, .

The Quadrilateral  is shown below with its diagonals  and .

. We call the point of intersection :

The diagonals of a quadrilateral with two pairs of adjacent congruent sides - a kite - are perpendicular; also,  bisects the  and angles of the kite. Consequently,  is a 30-60-90 triangle and  is a 45-45-90 triangle. Also, the diagonal that connects the common vertices of the pairs of adjacent sides bisects the other diagonal, making  the midpoint of . Therefore, 

.

By the 30-60-90 Theorem, since  and  are the short and long legs of , 

By the 45-45-90 Theorem, since  and  are the legs of a 45-45-90 Theorem, 

.

The diagonal  has length 

.

In order for two rectangles to be similar, the ratio of their dimensions must be equal. We can check which dimensions are those of a rectangle similar to the given one by first calculating the ratio of the length to the width for the given rectangle, and then doing the same for each of the answer choices until we find which has an equal ratio between its dimensions:

So in order for a rectangle to be similar to the given rectangle, this must be the ratio of its length to its width. Now we check the answer choices, in no particular order, for one with this ratio:

We can see that only the rectangle with a length of    and a width of    has the same ratio as the given rectangle, so this is the similar one.