A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle  and use the Pythagorean Theorem to solve for . From the problem:

Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

Factoring,

 and 

The first solution is nonsensical for this problem. 

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle  to find the length of side . From the problem, we are given  and . Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

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, .

Write the formula for the area of a rhombus.

Plug in the given area and diagonal length. Solve for the other diagonal.

Using the Law of Sines,

1) All sides of a rhombus are congruent.

2) Because all sides of a rhombus are congruent, the expressions of the side lengths can be set equal to each other.  The resulting equation is then solved,

3) Because the sides of a rhombus are congruent,  can be substituted into either  or  to find the length of a side,

, or, .

4) Each of the composing triangles are right triangles, so then  is the length of the hypotenuse for each triangle.

5) .

6) The standard  right triangle has a hypotenuse length equal to .

7) The hypotenuse of a standard  right triangle is being multiplied by .  

The result is , so then  is the scale factor for the triangle side lengths.

8) For the standard  right triangle, the other two side lengths are  and , so then the height of the triangle from step 7) has a height of , and the base length is .

9) The base of the triangle from step 7) is 

,

and the height is 

.

10) Diagonal 

,

and diagonal 

.

Because a rhombus has vertical and horizontal symmetry, it can be broken into four congruent triangles, each with a hypotenuse of 13 and a base of 5 (half the given diagonal).

The Pythagorean Theorem

 will yield,

  for the height of the triangles.

The greater diagonal is twice the height of the triangles therefore, the greater diagonal becomes:

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle  to find the length of diagonal . From the given information, each of the sides of the rhombus measures  meters and .

Because the diagonals bisect each other, we know:

Using the Pythagorean theorem,