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Advanced Geometry : Rhombuses

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Example Questions

Example Question #1 : Rhombuses

Which of the following shapes is a rhombus?

Shapes

Possible Answers:

None

A, B

B, D

Correct answer:

Explanation:

A rhombus is a four-sided figure where all sides are straight and equal in length. All opposite sides are parallel. A square is considered to be a rhombus.

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Example Question #2 : How To Find The Area Of A Rhombus

Assume quadrilateral MNOP is a rhombus. If diagonal $\overline{MO}=29$ and diagonal $\overline{NP}=14$ , what is the area of rhombus MNOP

Possible Answers:

203

32.2

406

101.5

20.15

Correct answer:

203

Explanation:

Solving for the area of rhombus MNOP requires knowledge of the equation for finding the area of a rhombus. The equation is $\small A=\frac{pq}{2}$ , where and $\small q$ are the two diagonals of the rhombus. Since both of these values are given to us in the original problem, we merely need to substitute these values into the equation to obtain:

$\small A=\frac{(29)(14)}{2}=\frac{406}{2}=203$

The area of rhombus $\small MNOP$ is therefore $\small 203$ square units.

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Example Question #3 : Rhombuses

Screen_shot_2015-03-06_at_3.03.05_pm

What is the area of the rhombus above?

Possible Answers:

108

Correct answer:

Explanation:

The formula for the area of a rhombus from the diagonals is half the product of the diagonals, or in mathematical terms:

$A=\frac{p\cdot q}{2}$ where and are the lengths of the diagonals.

Substituting our values yields,

$A=\frac{8*12}{2}=\frac{96}{2}=48$

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Example Question #91 : Quadrilaterals

Screen_shot_2015-03-06_at_5.54.17_pm

Above is a rhombus imposed on a rectangle. What is the area of the rhombus?

Possible Answers:

Correct answer:

Explanation:

One of the formulas for a rhombus is base times height,

$A=b \cdot h$

Since a rhombas has equal sides, the base is 5 and the height of the rhombus is the same as the height of the rectangle, 4.

Substituting in these values we get the following:

$A=5 \cdot 4 =20$

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Example Question #92 : Quadrilaterals

Assume quadrilateral $\small ABCD$ is a rhombus. The perimeter of $\small ABCD$ is , and the length of one of its diagonals is $\small 12$ . What is the area of $\small ABCD$ ?

Possible Answers:

$\small 96$

$\small 48$

$\small 64$

$\small 100$

$\small 36$

Correct answer:

$\small 96$

Explanation:

To solve for the area of the rhombus $\small ABCD$ , we must use the equation $\small A=\frac{pq}{2}$ , where $\small p$ and $\small q$ are the diagonals of the rhombus. Since the perimeter of the rhombus is $\small 40$ , and by definition all 4 sides of a rhombus have the same length, we know that the length of each side is $\small 10$ . We can find the length of the other diagonal if we recognize that the two diagonals combined with a side edge form a right triangle. The length of the hypotenuse is $\small 10$ , and each leg of the triangle is equal to one-half of each diagonal. We can therefore set up an equation involving Pythagorean's Theorem as follows:

$\small 10^2=x^2+6^2$ , where $\small x$ is equal to one-half the length of the unknown diagonal.

We can therefore solve for $\small x$ as follows:

$\small x^2=10^2-6^2=100-36=64$

$\small x$ is therefore equal to 8, and our other diagonal is 16. We can now use both diagonals to solve for the area of the rhombus:

$\small A=\frac{(16)(12)}{2}=\frac{192}{2}=96$

The area of rhombus $\small ABCD$ is therefore equal to $\small 96$

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Example Question #2 : How To Find The Area Of A Rhombus

What is the area of a rhombus with diagonal lengths of $\sqrt2$ and $\sqrt3$ ?

Possible Answers:

$\sqrt6$

$\frac{\sqrt6}{2}$

$2\sqrt2$

$2\sqrt3$

$\frac{\sqrt3}{2}$

Correct answer:

$\frac{\sqrt6}{2}$

Explanation:

The area of a rhombus is given below. Plug in the diagonals and solve for the area.

$A=\frac{d1 \cdot d2}{2} = \frac{\sqrt2 \cdot \sqrt3}{2}=\frac{\sqrt6}{2}$

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Example Question #2 : Rhombuses

Find the area of a rhombus if the diagonal lengths are and .

Possible Answers:

$\frac{1}{2}$

$\sqrt2$

$2\sqrt2$

Correct answer:

Explanation:

The area of the rhombus is given below. Substitute the diagonals into the formula.

$A=\frac{d1\cdot d2}{2} = \frac{(1)(2)}{2}=1$

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Example Question #1 : How To Find The Area Of A Rhombus

Find the area of a rhombus if its diagonal lengths are $\sqrt{2x}\:cm$ and $\sqrt{4x}\:cm$ .

Possible Answers:

$4x\:cm^2$

$2\sqrt{2x}\:cm^2$

$8x\:cm^2$

$2x\sqrt2\:cm^2$

$x\sqrt2\:cm^2$

Correct answer:

$x\sqrt2\:cm^2$

Explanation:

Write the equation for the area of a rhombus.

$A= \frac{d_1 \cdot d_2}{2}$

Substitute the diagonals and evaluate the area.

$A= \frac{\sqrt{2x}\:cm\sqrt{4x}\:cm}{2} = \frac{\sqrt{8x^2}}{2}\:cm^2 = \frac{2x\sqrt2}{2}\:cm^2=x\sqrt2\:cm^2$

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Example Question #1 : How To Find The Area Of A Rhombus

Find the area of a rhombus if the both diagonals have a length of $\sqrt{x+2}$ .

Possible Answers:

$\frac{\sqrt{2x+4}}{2}$

2x+4

x+2

$\sqrt{2x+4}$

$\frac{x+2}{2}$

Correct answer:

$\frac{x+2}{2}$

Explanation:

Write the formula for the area of a rhombus.

$A=\frac{d_1 \cdot d_2}{2}$

Since both diagonals are equal, d_1=d_2 . Plug in the diagonals and reduce.

$A=\frac{(\sqrt x+2)(\sqrt x+2)}{2}=\frac{\sqrt{x+2}^2}{2}= \frac{x+2}{2}$

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Example Question #1 : How To Find The Area Of A Rhombus

What is the area of a rhombus if the diagonals are $\frac{6}{x}$ and $\frac{6}{x^2}$ ?

Possible Answers:

$\frac{18}{x^3}$

$\frac{36}{x^3}$

$\frac{6\sqrt x}{x}$

$\frac{36}{x^2}$

$\frac{18}{x^2}$

Correct answer:

$\frac{18}{x^3}$

Explanation:

Write the formula for an area of a rhombus.

$A=\frac{d_1\cdot d_2}{2}$

Substitute the diagonal lengths provided into the formula.

$A=\frac{d1\cdot d2}{2} = \frac{\frac{6}{x}\cdot\frac{6}{x^2}}{2}$

Multiply the two terms in the numerator.

$A= \frac{\frac{36}{x^3}}{2}$

You can consider the outermost division by two as multiplying everything in the numerator by $\frac{1}{2}$ .

$A= \frac{1}{2}(\frac{36}{x^3})$

Multiply across and reduce to arrive at the correct answer.

$A= \frac{1}{2}(\frac{36}{x^3})=\frac{36}{2x^3}=\frac{18}{x^3}$

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Advanced Geometry : Rhombuses

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #1 : Rhombuses

Example Question #2 : How To Find The Area Of A Rhombus

Example Question #3 : Rhombuses

Example Question #91 : Quadrilaterals

Example Question #92 : Quadrilaterals

Example Question #2 : How To Find The Area Of A Rhombus

Example Question #2 : Rhombuses

Example Question #1 : How To Find The Area Of A Rhombus

Example Question #1 : How To Find The Area Of A Rhombus

Example Question #1 : How To Find The Area Of A Rhombus

All Advanced Geometry Resources

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