Geometry - Rhombuses | Practice Hub

Given: Rhombuses and .

$m \angle A = 60 ^{\circ }$ and $m \angle F = 120 ^{\circ }$

True, false, or undetermined: Rhombus $ABCD \sim$ Rhombus .

True

False

Undetermined

Explanation

Two figures are similar by definition if all of their corresponding sides are proportional and all of their corresponding angles are congruent.

By definition, a rhombus has four sides that are congruent. If we let $s_{1}$ be the common sidelength of Rhombus and $s_{2}$ be the common sidelength of Rhombus , it can easily be seen that the ratio of the length of each side of the former to that of the latter is the same ratio, namely, $\frac{s_{1}}{s_{2}}$ .

Also, a rhombus being a parallelogram, its opposite angles are congruent, and its consecutive angles are supplementary. Therefore, since $m \angle A = 60 ^{\circ }$ , it follows that $m\angle C = 60 ^{\circ }$ , and $m \angle B = m \angle D = 180 ^{\circ } - 60^{\circ } = 120 ^{\circ }$ . By a similar argument, $m \angle H = m \angle F = 120 ^{\circ }$ and $m \angle E = m \angle G= 180 ^{\circ } - 120 ^{\circ } = 60^{\circ }$ . Therefore,

$\angle A \cong \angle E$

$\angle B \cong \angle F$

$\angle C \cong \angle F$

$\angle D \cong \angle H$

Since all corresponding sides are proportional and all corresponding angles are congruent, it holds that Rhombus $ABCD \sim$ Rhombus .

Find the perimeter of a rhombus if it has an area of and a diagonal length of .

Explanation

Recall the following properties of a rhombus, as shown by the figure above: The diagonals of a rhombus are perpendicular and they bisect each other. Thus, if we know the lengths of the diagonals, we can use the Pythagorean theorem to find the length of a side of the rhombus.

Recall how to find the area of a rhombus:

$\text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(1200)}{40}=60$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\text{Half diagonal 1}=\frac{40}{2}=20$

$\text{Half diagonal 2}=\frac{60}{2}=30$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\text{Side of Rhombus}=\sqrt{20^2+30^2}=\sqrt{1300}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\text{Perimeter of rhombus}=4\sqrt{1300}=144.22$

Make sure to round to places after the decimal.

Find the perimeter of a rhombus if it has an area of and a diagonal length of .

Explanation

Recall how to find the area of a rhombus:

$\text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(1200)}{40}=60$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\text{Half diagonal 1}=\frac{40}{2}=20$

$\text{Half diagonal 2}=\frac{60}{2}=30$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\text{Side of Rhombus}=\sqrt{20^2+30^2}=\sqrt{1300}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\text{Perimeter of rhombus}=4\sqrt{1300}=144.22$

Make sure to round to places after the decimal.

A rhombus has a side length of $\frac{5}{6}$ foot, what is the length of the perimeter (in inches).

inches

feet

inches

Explanation

To find the perimeter, first convert $\frac{5}{6}$ foot into the equivalent amount of inches. Since, $\frac{5}{6}=\frac{10}{12}$ and $\frac{12}{12}=1foot$ , $\frac{5}{6}$ is equal to inches.

Then apply the formula , where is equal to the length of one side of the rhombus.

Since,

The solution is:

Given that a rhombus has a perimeter of , find the length of one side of the rhombus.

Explanation

The perimeter of a rhombus is equal to , where the length of one side of the rhombus.

Since , we can set up the following equation and solve for .

$124=4\times (S)$

$S=\frac{124}{4}=31$

Find the length of a side of a rhombus that has diagonal lengths of and .

Explanation

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

$\text{Half Diagonal 1}=\frac{12}{2}=6$

$\text{Half Diagonal 2}=\frac{14}{2}=7$

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

$\text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}$

Plug in the lengths of the half diagonals to find the length of the rhombus.

$\text{Side of Rhombus}=\sqrt{6^2+7^2}=\sqrt{85}=9.22$

Make sure to round to places after the decimal.

Find the length of a side of a rhombus that has diagonal lengths of and .

Explanation

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

$\text{Half Diagonal 1}=\frac{12}{2}=6$

$\text{Half Diagonal 2}=\frac{14}{2}=7$

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

$\text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}$

Plug in the lengths of the half diagonals to find the length of the rhombus.

$\text{Side of Rhombus}=\sqrt{6^2+7^2}=\sqrt{85}=9.22$

Make sure to round to places after the decimal.

Find the area of the rhombus.

Explanation

Recall that one of the ways to find the area of a rhombus is with the following formula:

$\text{Area}=\text{base}\times\text{height}$

Now, since all four sides in the rhombus are the same, we know from the given side value what the length of the base will be. In order to find the length of the height, we will need to use sine.

$\sin x=\frac{\text{height}}{\text{side}}$ , where is the given angle.