## Example Questions

### Example Question #54 : Rhombuses

is a rhombus.
,
, and
. Find
.

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.

Example Question #11 : How To Find The Length Of The Diagonal Of A Rhombus

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

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 #12 : How To Find The Length Of The Diagonal Of A Rhombus

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.

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 #13 : How To Find The Length Of The Diagonal Of A Rhombus

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

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 #14 : How To Find The Length Of The Diagonal Of A Rhombus

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

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 #15 : How To Find The Length Of The Diagonal Of A Rhombus

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?

Write the formula for the area of a rhombus.

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

Example Question #16 : How To Find The Length Of The Diagonal Of A Rhombus

is a rhombus. Find
.

Using the Law of Sines,

Example Question #17 : How To Find The Length Of The Diagonal Of A Rhombus

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

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 #18 : How To Find The Length Of The Diagonal Of A Rhombus

What is the second diagonal for the above rhombus?

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 #19 : How To Find The Length Of The Diagonal Of A Rhombus

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

12

24

15

30

5

24

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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