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Advanced Geometry : Advanced Geometry

Study concepts, example questions & explanations for Advanced Geometry

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6 Diagnostic Tests 57 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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

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

Possible Answers:

$2\sqrt3$

$2\sqrt2$

$\frac{\sqrt6}{2}$

$\frac{\sqrt3}{2}$

$\sqrt6$

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 #1 : Rhombuses

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

Possible Answers:

$2\sqrt2$

$\frac{1}{2}$

$\sqrt2$

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 #2 : 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:

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

$8x\:cm^2$

$2x\sqrt2\:cm^2$

$4x\:cm^2$

$x\sqrt2\:cm^2$

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 #6 : Rhombuses

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

Possible Answers:

$\sqrt{2x+4}$

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

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

x+2

2x+4

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 #2 : 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:

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

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

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

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

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

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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Example Question #11 : Rhombuses

Find the area of a rhombus with diagonal lengths of $(2x+2)\:in$ and $(4x+4)\:in$ .

Possible Answers:

$(8x+6)\:in^2$

$(3x+3)\:in^2$

$(4x^2+8x+4)\:in^2$

$(4x+3)\:in^2$

$(8x^2+16x+8)\:in^2$

Correct answer:

$(4x^2+8x+4)\:in^2$

Explanation:

Write the formula for the area of a rhombus.

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

Substitute the given diagonal lengths:

$A=\frac{(2x+2)(4x+4)}{2}$

Use FOIL to multiply the two parentheticals in the numerator:

First: $2x\cdot4x=8x^2$

Outer: $2x\cdot4=8x$

Inner: $2\cdot4x=8x$

Last: $2\cdot 4=8$

Add your results together:

$A=\frac{8x^2+16x+8}{2}$

Divide all elements in the numerator by two to arrive at the correct answer:

$A=(4x^2+8x+4) \:in^2$

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

Rhombus_1

The above figure shows a rhombus RHOM . Give its area.

Possible Answers:

120

$40\sqrt{61}$

$40 \sqrt{11}$

Correct answer:

$40 \sqrt{11}$

Explanation:

Construct the other diagonal of the rhombus, which, along with the first one, form a pair of mutual perpendicular bisectors.

Rhombus_1

By the Pythagorean Theorem,

$n = \sqrt{12^{2}- 10^{2}} =\sqrt{144- 100} = \sqrt{44}= \sqrt{4}\cdot \sqrt{11} = 2 \sqrt{11}$

The rhombus can be seen as the composite of four congruent right triangles, each with legs 10 and $2 \sqrt{11}$ , so the area of the rhombus is

$A = 4 \cdot \frac{1}{2}\cdot 10 \cdot 2\sqrt{11} =40 \sqrt{11}$ .

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Example Question #201 : Geometry

Rhombus ABCD has perimeter 48; AC = 12 . What is the area of Rhombus ABCD ?

Possible Answers:

144

$72 \sqrt{3}$

$144 \sqrt{3}$

$72 \sqrt{2}$

Correct answer:

$72 \sqrt{3}$

Explanation:

Each side of a rhombus is congruent, so if it has perimeter 48, it has sidelength 12. Also, the diagonals of a rhombus are each other's perpendicular bisectors, so if they are both constructed, and their point of intersection is called , then AM = 6 . The following figure is formed by the rhombus and its diagonals.

Untitled

$\bigtriangleup AMB$ is a right triangle with its short leg half the length of its hypotenuse, so it is a 30-60-90 triangle, and its long leg measures $6\sqrt{3}$ by the 30-60-90 Theorem. Therefore, $BD = 12\sqrt{3}$ . The area of a rhombus is half the product of the lengths of its diagonals:

$\frac{1}{2} \times 12 \times 12 \sqrt{3} = 72 \sqrt{3}$

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Example Question #11 : Rhombuses

A rhombus contains diagonals with the length $12 \:cm$ and $6 \:cm$ . Find the area of the rhombus.

Possible Answers:

$72\:cm^2$

$54\:cm^2$

$96\:cm^2$

$88\:cm^2$

$36\:cm^2$

Correct answer:

$36\:cm^2$

Explanation:

The equation for the area of a rhombus is given by:

$A= \frac{p \cdot q}{2}$

where and are the two diagonal lengths.

This problem very quickly becomes one of the "plug and chug" type, where the given values just need to be substituted into the equation and the equation then solved. By plugging in the values given, we get:

$A = \frac{12\:cm \cdot 6\:cm}{2}$

$A = \frac{72\:cm^2}2{}$

$A= 36\:cm^2$

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

Find the area of a rhombus if the diagonals lengths are $20\:cm$ and $40\:cm$ .

Possible Answers:

$900\:cm^2$

$600\:cm^2$

$200\:cm^2$

$800\:cm^2$

$400\:cm^2$

Correct answer:

$400\:cm^2$

Explanation:

Write the formula for the area of a rhombus:

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

Substitute the given lengths of the diagonals and solve:

$A=\frac{d1 \cdot d2}{2} = \frac{20\:cm \cdot 40\:cm}{2} =\frac{800\:cm^2}{2} = 400\:cm^2$

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Advanced Geometry : Advanced Geometry

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

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

Example Question #1 : Rhombuses

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

Example Question #6 : Rhombuses

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

Example Question #11 : Rhombuses

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

Example Question #201 : Geometry

Example Question #11 : Rhombuses

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

All Advanced Geometry Resources

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