To find the value of diagonal, we must first recognize some important properties of rhombuses. Since the perimeter is ofis, and by definition a rhombus has four sides of equal length, each side length of the rhombus is equal to. The diagonals of rhombuses also form four right triangles, with hypotenuses equal to the side length of the rhombus and legs equal to one-half the lengths of the diagonals. We can therefore use the Pythagorean Theorem to solve for one-half of the unknown diagonal:

, whereis the rhombus side length,is one-half of the known diagonal, andis one-half of the unknown diagonal. We can therefore solve for:

is therefore equal to. Sincerepresents one-half of the unknown diagonal, we need to multiply byto find the full length of diagonal.

The length of diagonalis therefore

This problem relies on the knowledge of the equation for the area of a rhombus,, whereis the area, andandare the lengths of the individual diagonals. We can substitute the values that we know into the equation to obtain:

Therefore, our final answer is that the diagonal

The area of a rhombus is given below.

Substitute the given area and a diagonal. Solve for the other diagonal.

The area of a rhombus is given below. Plug in the area and the given diagonal. Solve for the other diagonal.

Let the shorter diagonal be, and the longer diagonal be. The longer dimension is twice as long as the other diagonal. Write an expression for this.

Write the area of the rhombus.

Since we are solving for the shorter diagonal, it's best to setup the equation in terms, so that we can solve for the shorter diagonal. Plug in the area and expression to solve for.

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 triangleto find the length of diagonal. From the problem, we are given that the sides areand. Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

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 triangleto find the length of diagonal. From the problem, we are given that the sides areand. Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

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 triangleto find the length of diagonal. From the problem, we are given that the sides areand. Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

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 triangleand use the Pythagorean Theorem to solve for. From the problem:

Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

Using the quadratic formula,

With this equation, we get two solutions:

Only the positive solution is valid 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 triangleand 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.