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Example Questions
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 #11 : 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 #12 : 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 #12 : 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 #141 : 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?
Write the formula for the area of a rhombus.
Plug in the given area and diagonal length. Solve for the other diagonal.
Example Question #51 : Rhombuses
is a rhombus. Find
.
Using the Law of Sines,
Example Question #11 : 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 #11 : 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 #13 : 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
?
24
12
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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