# Advanced Geometry : How to find the length of the diagonal of a rhombus

## Example Questions

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### Example Question #141 : Quadrilaterals

is a rhombus.  and . Find the length of the sides.

Explanation:

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,

### Example Question #7 : Calculating The Length Of The Diagonal Of A Quadrilateral

Rhombus  has area 56.

Which of the following could be true about the values of  and ?

None of the other responses gives a correct answer.

Explanation:

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.

### Example Question #8 : Calculating The Length Of The Diagonal Of A Quadrilateral

Rhombus  has perimeter 64; . What is the length of  ?

Explanation:

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, .

### Example Question #9 : Calculating The Length Of The Diagonal Of A Quadrilateral

Rhombus  has perimeter 48; . What is the length of  ?

Explanation:

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, .

### Example Question #142 : Quadrilaterals

If the area of a rhombus is , and the length of one of its diagonals is , what must be the length of the other diagonal?

Explanation:

Write the formula for the area of a rhombus.

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

### Example Question #141 : Plane Geometry

is a rhombus.  Find .

Explanation:

Using the Law of Sines,

### Example Question #141 : Plane Geometry

Find the lengths of the two diagonals, the longer diagonal is , the shorter diagonal is .

Explanation:

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

.

### Example Question #145 : Quadrilaterals

What is the second diagonal for the above rhombus?

Explanation:

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:

### Example Question #146 : Quadrilaterals

is rhombus with side lengths in meters.  and . What is the length, in meters, of ?

24

30

5

12

15

24

Explanation:

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,

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