Rhombuses - ISEE Upper Level: Quantitative Reasoning

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Question

In Rhombus , and . Which is the greater quantity?

(A)

(B)

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Answer

The four sides of a rhombus, by defintion, have equal length, so

$7x\div 7 = 8y \div 7$

$x = \frac{8}{7}y$

Since and are positive, .

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Question

Consider the rhombus below. Solve for .

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Answer

The total sum of the interior angles of a quadrilateral is degrees. In this problem, we are only considering half of the interior angles:

$\frac{360}{2}=180$

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Question

Note: Figure NOT drawn to scale.

The above depicts a rhombus and one of its diagonals. What is ?

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Answer

The diagonals of a rhombus bisect the angles.

The angle bisected must be supplementary to the $122^{\circ }$ angle since they are consecutive angles of a parallelogram; therefore, that angle has measure $\left (180 - 122 \right ) ^{\circ } = 58^{\circ }$ , and is half that, or $29^{\circ }$ .

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Question

Two diagonals of a rhombus have lengths of and . Give the area in terms of .

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Answer

The formula for the area of a rhombus is

$Area=\frac{d_{1}d_{2}}{2}$ ,

where is the length of one diagonal and is the length of the other diagonal.

$Area=\frac{d_{1}d_{2}}{2}=\frac{(t-1)(t+1)}{2}=\frac{t^2-1}{2}$

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Question

A rhombus has side length $4\ cm$ , and one of the interior angles is $30^{\circ}$ . Give the area of the rhombus.

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Answer

The area of a rhombus can be determined by the following formula:

$Area=s^2sin\alpha$ ,

where is the length of any side and $\alpha$ is any interior angle.

$Area=s^2sin\alpha=4^2\times sin30^{\circ}=16\times \frac{1}{2}=8\ cm^2$

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Question

Find the area of a rhombus with one diagonal having a length of 12in and the other having a length two times the first diagonal.

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Answer

To find the area of a rhombus, we will use the following formula:

$A = \frac{pq}{2}$

where p and q are the lengths of the diagonals of the rhombus.

Now, we know one diagonal has a length of 12in. We also know the other diagonal is two times the first diagonal. Therefore, the second diagonal is 24in.

Knowing this, we can substitute into the formula. We get

$A = \frac{12\text{in} \cdot 24\text{in}}{2}$

$A = \frac{288\text{in}^2}{2}$

$A = 144\text{in}^2$

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Question

A rhombus has a perimeter of $36\ in$ . Give the side length of the rhombus.

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Answer

Like any polygon, the perimeter of the rhombus is the total distance around the outside, which can be found by adding together the length of each side. In the case of a rhombus, all four sides have the same length, i.e.

,

where is the length of each side.

$Perimeter=4a\Rightarrow 36=4a\Rightarrow a=9\ in$

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Question

A rhombus has the area of $9\sqrt{2}\ cm^2$ . One of the interior angles is $45^{\circ}$ . Give the length of each side of the rhombus.

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Answer

The area of a rhombus can be determined by the following formula:

$Area=s^2sin\alpha$ ,

where is the length of any side and $\alpha$ is any interior angle.

$Area=s^2sin\alpha\Rightarrow 9\sqrt{2}=s^2\times sin45^{\circ}\Rightarrow 9\sqrt{2}=s^2 \times \frac{\sqrt{2}}{2}$

$\Rightarrow s^2=18\Rightarrow s=\sqrt{18}=\sqrt{9\times 2}=3\sqrt{2}\ cm$

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Question

One side of a rhombus measures 2 yards. Which of the following is equal to the perimeter?

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Answer

A rhombus has four sides of equal measure, so the perimeter of a rhombus with sidelength 2 yards is $2 \times 4 = 8$ yards.

$8 \textrm{ yd} = 8 \times 3 = 24 \textrm{ ft} = 24 \times 12 = 288 \textrm{ in}$

Also,

$8 \textrm{ yd} = 24 \textrm{ ft} =\frac{ 24}{5,280} = \frac{1}{220} \textrm{ mi}$

The only choice that agrees with any of these measures is 288 inches, the correct choice.

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Question

One side of a rhombus measures . Give its perimeter in terms of .

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Answer

A rhombus has four sides of equal measure, so the perimeter of a rhombus with sidelength is .

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Question

In Rhombus , and . Which is the greater quantity?

(A)

(B)

Tap to reveal answer

Answer

The four sides of a rhombus, by defintion, have equal length, so

$7x\div 7 = 8y \div 7$

$x = \frac{8}{7}y$

Since and are positive, .

← Didn't Know|Knew It →

Question

Consider the rhombus below. Solve for .

Tap to reveal answer

Answer

The total sum of the interior angles of a quadrilateral is degrees. In this problem, we are only considering half of the interior angles:

$\frac{360}{2}=180$

← Didn't Know|Knew It →

Question

Note: Figure NOT drawn to scale.

The above depicts a rhombus and one of its diagonals. What is ?

Tap to reveal answer

Answer

The diagonals of a rhombus bisect the angles.

The angle bisected must be supplementary to the $122^{\circ }$ angle since they are consecutive angles of a parallelogram; therefore, that angle has measure $\left (180 - 122 \right ) ^{\circ } = 58^{\circ }$ , and is half that, or $29^{\circ }$ .

← Didn't Know|Knew It →

Question

Two diagonals of a rhombus have lengths of and . Give the area in terms of .

Tap to reveal answer

Answer

The formula for the area of a rhombus is

$Area=\frac{d_{1}d_{2}}{2}$ ,

where is the length of one diagonal and is the length of the other diagonal.

$Area=\frac{d_{1}d_{2}}{2}=\frac{(t-1)(t+1)}{2}=\frac{t^2-1}{2}$

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Question

A rhombus has side length $4\ cm$ , and one of the interior angles is $30^{\circ}$ . Give the area of the rhombus.

Tap to reveal answer

Answer

The area of a rhombus can be determined by the following formula:

$Area=s^2sin\alpha$ ,

where is the length of any side and $\alpha$ is any interior angle.

$Area=s^2sin\alpha=4^2\times sin30^{\circ}=16\times \frac{1}{2}=8\ cm^2$

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Question

Find the area of a rhombus with one diagonal having a length of 12in and the other having a length two times the first diagonal.

Tap to reveal answer

Answer

To find the area of a rhombus, we will use the following formula:

$A = \frac{pq}{2}$

where p and q are the lengths of the diagonals of the rhombus.

Now, we know one diagonal has a length of 12in. We also know the other diagonal is two times the first diagonal. Therefore, the second diagonal is 24in.

Knowing this, we can substitute into the formula. We get

$A = \frac{12\text{in} \cdot 24\text{in}}{2}$

$A = \frac{288\text{in}^2}{2}$

$A = 144\text{in}^2$

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Question

A rhombus has a perimeter of $36\ in$ . Give the side length of the rhombus.

Tap to reveal answer

Answer

Like any polygon, the perimeter of the rhombus is the total distance around the outside, which can be found by adding together the length of each side. In the case of a rhombus, all four sides have the same length, i.e.

,

where is the length of each side.

$Perimeter=4a\Rightarrow 36=4a\Rightarrow a=9\ in$

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Question

A rhombus has the area of $9\sqrt{2}\ cm^2$ . One of the interior angles is $45^{\circ}$ . Give the length of each side of the rhombus.

Tap to reveal answer

Answer

The area of a rhombus can be determined by the following formula:

$Area=s^2sin\alpha$ ,

where is the length of any side and $\alpha$ is any interior angle.

$Area=s^2sin\alpha\Rightarrow 9\sqrt{2}=s^2\times sin45^{\circ}\Rightarrow 9\sqrt{2}=s^2 \times \frac{\sqrt{2}}{2}$

$\Rightarrow s^2=18\Rightarrow s=\sqrt{18}=\sqrt{9\times 2}=3\sqrt{2}\ cm$

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Question

One side of a rhombus measures 2 yards. Which of the following is equal to the perimeter?

Tap to reveal answer

Answer

A rhombus has four sides of equal measure, so the perimeter of a rhombus with sidelength 2 yards is $2 \times 4 = 8$ yards.

$8 \textrm{ yd} = 8 \times 3 = 24 \textrm{ ft} = 24 \times 12 = 288 \textrm{ in}$

Also,

$8 \textrm{ yd} = 24 \textrm{ ft} =\frac{ 24}{5,280} = \frac{1}{220} \textrm{ mi}$

The only choice that agrees with any of these measures is 288 inches, the correct choice.

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Question

One side of a rhombus measures . Give its perimeter in terms of .

Tap to reveal answer

Answer

A rhombus has four sides of equal measure, so the perimeter of a rhombus with sidelength is .

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