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Advanced Geometry : Plane 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 #21 : Rhombuses

Find the area of the rhombus below.

Possible Answers:

5.20

14.42

21.09

4.97

Correct answer:

4.97

Explanation:

Recall that the diagonals of the rhombus are perpendicular bisectors. From the given side and the given diagonal, we can find the length of the second diagonal by using the Pythagorean Theorem.

$\text{side}^2=(\frac{\text{diagonal 1}}{2})^2-(\frac{\text{diagonal 2}}{2})^2$

Let the given diagonal be diagonal 1, and rearrange the equation to solve for diagonal 2.

$(\frac{\text{diagonal 2}}{2})^2=\text{side}^2-(\frac{\text{diagonal 1}}{2})^2$

$(\frac{\text{diagonal 2}}{2})=\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

$\text{diagonal 2}=2\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

Plug in the given side and diagonal to find the length of diagonal 2.

$\text{diagonal 2}=2\sqrt{5^2-(\frac{1}{2})^2}=2\sqrt{\frac{99}{4}}$

Now, recall how to find the area of a rhombus:

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

Plug in the two diagonals to find the area.

$\text{Area of Rhombus}=\frac{(1)(2\sqrt{\frac{99}{4}})}{2}=4.97$

Make sure to round to places after the decimal.

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Example Question #403 : Advanced Geometry

Find the area of the rhombus below.

Possible Answers:

34.42

20.40

41.28

23.92

Correct answer:

23.92

Explanation:

Recall that the diagonals of the rhombus are perpendicular bisectors. From the given side and the given diagonal, we can find the length of the second diagonal by using the Pythagorean Theorem.

$\text{side}^2=(\frac{\text{diagonal 1}}{2})^2-(\frac{\text{diagonal 2}}{2})^2$

Let the given diagonal be diagonal 1, and rearrange the equation to solve for diagonal 2.

$(\frac{\text{diagonal 2}}{2})^2=\text{side}^2-(\frac{\text{diagonal 1}}{2})^2$

$(\frac{\text{diagonal 2}}{2})=\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

$\text{diagonal 2}=2\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

Plug in the given side and diagonal to find the length of diagonal 2.

$\text{diagonal 2}=2\sqrt{12^2-(\frac{2}{2})^2}=2\sqrt{143}$

Now, recall how to find the area of a rhombus:

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

Plug in the two diagonals to find the area.

$\text{Area of Rhombus}=\frac{(2)(2\sqrt{143})}{2}=23.92$

Make sure to round to places after the decimal.

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Example Question #111 : Plane Geometry

Find the area of the rhombus below.

Possible Answers:

39.19

35.51

36.02

24.19

Correct answer:

39.19

Explanation:

Recall that the diagonals of the rhombus are perpendicular bisectors. From the given side and the given diagonal, we can find the length of the second diagonal by using the Pythagorean Theorem.

$\text{side}^2=(\frac{\text{diagonal 1}}{2})^2-(\frac{\text{diagonal 2}}{2})^2$

Let the given diagonal be diagonal 1, and rearrange the equation to solve for diagonal 2.

$(\frac{\text{diagonal 2}}{2})^2=\text{side}^2-(\frac{\text{diagonal 1}}{2})^2$

$(\frac{\text{diagonal 2}}{2})=\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

$\text{diagonal 2}=2\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

Plug in the given side and diagonal to find the length of diagonal 2.

$\text{diagonal 2}=2\sqrt{10^2-(\frac{4}{2})^2}=2\sqrt{96}$

Now, recall how to find the area of a rhombus:

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

Plug in the two diagonals to find the area.

$\text{Area of Rhombus}=\frac{(4)(2\sqrt{96})}{2}=39.19$

Make sure to round to places after the decimal.

Report an Error

Example Question #112 : Plane Geometry

Find the area of the rhombus below.

Possible Answers:

32.69

30.09

26.46

28.55

Correct answer:

32.69

Explanation:

Recall that the diagonals of the rhombus are perpendicular bisectors. From the given side and the given diagonal, we can find the length of the second diagonal by using the Pythagorean Theorem.

$\text{side}^2=(\frac{\text{diagonal 1}}{2})^2-(\frac{\text{diagonal 2}}{2})^2$

Let the given diagonal be diagonal 1, and rearrange the equation to solve for diagonal 2.

$(\frac{\text{diagonal 2}}{2})^2=\text{side}^2-(\frac{\text{diagonal 1}}{2})^2$

$(\frac{\text{diagonal 2}}{2})=\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

$\text{diagonal 2}=2\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

Plug in the given side and diagonal to find the length of diagonal 2.

$\text{diagonal 2}=2\sqrt{11^2-(\frac{3}{2})^2}=2\sqrt{\frac{475}{4}}$

Now, recall how to find the area of a rhombus:

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

Plug in the two diagonals to find the area.

$\text{Area of Rhombus}=\frac{(3)(2\sqrt{\frac{475}{4}})}{2}=32.69$

Make sure to round to places after the decimal.

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Example Question #21 : Rhombuses

Find the area of the rhombus below.

Possible Answers:

77.02

83.45

79.58

82.05

Correct answer:

82.05

Explanation:

Recall that the diagonals of the rhombus are perpendicular bisectors. From the given side and the given diagonal, we can find the length of the second diagonal by using the Pythagorean Theorem.

$\text{side}^2=(\frac{\text{diagonal 1}}{2})^2-(\frac{\text{diagonal 2}}{2})^2$

Let the given diagonal be diagonal 1, and rearrange the equation to solve for diagonal 2.

$(\frac{\text{diagonal 2}}{2})^2=\text{side}^2-(\frac{\text{diagonal 1}}{2})^2$

$(\frac{\text{diagonal 2}}{2})=\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

$\text{diagonal 2}=2\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

Plug in the given side and diagonal to find the length of diagonal 2.

$\text{diagonal 2}=2\sqrt{14^2-(\frac{6}{2})^2}=2\sqrt{187}$

Now, recall how to find the area of a rhombus:

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

Plug in the two diagonals to find the area.

$\text{Area of Rhombus}=\frac{(6)(2\sqrt{187})}{2}=82.05$

Make sure to round to places after the decimal.

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Example Question #21 : How To Find The Area Of A Rhombus

Find the area of the rhombus below.

Possible Answers:

190.04

185.73

186.98

171.20

Correct answer:

185.73

Explanation:

Recall that the diagonals of the rhombus are perpendicular bisectors. From the given side and the given diagonal, we can find the length of the second diagonal by using the Pythagorean Theorem.

$\text{side}^2=(\frac{\text{diagonal 1}}{2})^2-(\frac{\text{diagonal 2}}{2})^2$

Let the given diagonal be diagonal 1, and rearrange the equation to solve for diagonal 2.

$(\frac{\text{diagonal 2}}{2})^2=\text{side}^2-(\frac{\text{diagonal 1}}{2})^2$

$(\frac{\text{diagonal 2}}{2})=\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

$\text{diagonal 2}=2\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

Plug in the given side and diagonal to find the length of diagonal 2.

$\text{diagonal 2}=2\sqrt{15^2-(\frac{14}{2})^2}=2\sqrt{176}$

Now, recall how to find the area of a rhombus:

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

Plug in the two diagonals to find the area.

$\text{Area of Rhombus}=\frac{(14)(2\sqrt{176})}{2}=185.73$

Make sure to round to places after the decimal.

Report an Error

Example Question #115 : Plane Geometry

Find the area of the rhombus below.

Possible Answers:

101.07

98.55

102.49

109.77

Correct answer:

109.77

Explanation:

Recall that the diagonals of the rhombus are perpendicular bisectors. From the given side and the given diagonal, we can find the length of the second diagonal by using the Pythagorean Theorem.

$\text{side}^2=(\frac{\text{diagonal 1}}{2})^2-(\frac{\text{diagonal 2}}{2})^2$

Let the given diagonal be diagonal 1, and rearrange the equation to solve for diagonal 2.

$(\frac{\text{diagonal 2}}{2})^2=\text{side}^2-(\frac{\text{diagonal 1}}{2})^2$

$(\frac{\text{diagonal 2}}{2})=\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

$\text{diagonal 2}=2\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

Plug in the given side and diagonal to find the length of diagonal 2.

$\text{diagonal 2}=2\sqrt{13^2-(\frac{9}{2})^2}=2\sqrt{\frac{595}{4}}$

Now, recall how to find the area of a rhombus:

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

Plug in the two diagonals to find the area.

$\text{Area of Rhombus}=\frac{(9)(2\sqrt{\frac{595}{4}})}{2}=109.77$

Make sure to round to places after the decimal.

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Example Question #31 : How To Find The Area Of A Rhombus

Find the area of the rhombus below.

Possible Answers:

228.95

339.58

341.20

301.29

Correct answer:

228.95

Explanation:

Recall that the diagonals of the rhombus are perpendicular bisectors. From the given side and the given diagonal, we can find the length of the second diagonal by using the Pythagorean Theorem.

$\text{side}^2=(\frac{\text{diagonal 1}}{2})^2-(\frac{\text{diagonal 2}}{2})^2$

Let the given diagonal be diagonal 1, and rearrange the equation to solve for diagonal 2.

$(\frac{\text{diagonal 2}}{2})^2=\text{side}^2-(\frac{\text{diagonal 1}}{2})^2$

$(\frac{\text{diagonal 2}}{2})=\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

$\text{diagonal 2}=2\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

Plug in the given side and diagonal to find the length of diagonal 2.

$\text{diagonal 2}=2\sqrt{20^2-(\frac{12}{2})^2}=2\sqrt{364}$

Now, recall how to find the area of a rhombus:

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

Plug in the two diagonals to find the area.

$\text{Area of Rhombus}=\frac{(12)(2\sqrt{364})}{2}=228.95$

Make sure to round to places after the decimal.

Report an Error

Example Question #32 : Rhombuses

A rhombus is $d_{2} = 10$ meters across. $d_{1} = \frac{1}{2}d_{2}$ . Find the area.

Possible Answers:

m²

$12\textup{ m}^2$

$20\textup{ m}^2$

$25\textup{ m}^2$

$5\textup{ m}^2$

Correct answer:

$25\textup{ m}^2$

Explanation:

1) Substitute $d_{2} = 10$ into the expression $d_{1} = \frac{1}{2}d_{2}$ , $d_{1} = \frac{1}{2}\cdot 10 = 5$ .

2) Substitute $d_{2} = 10$ and $d_{1} = 5$ into the rhombus area formula,

$A = \frac{1}{2}d_{1}d_{2}$

$A = \frac{1}{2}\cdot 5 \cdot 10$

$= \frac{1}{2}\cdot 50$

$= 25\textup{ m}^2$

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Example Question #33 : Rhombuses

Find the area of the rhombus.

Possible Answers:

12.44

14.95

18.05

13.62

Correct answer:

13.62

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.

$\text{height}=(\text{side})(\sin x)$

Now, plug this into the equation for the area to get the following equation:

$\text{Area}=(side)^2 \sin x$

Plug in the given side length and angle values to find the area.

$\text{Area}=(5)^2\sin 33=13.62$

Make sure to round to places after the decimal.

Report an Error

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Advanced Geometry : Plane Geometry

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #21 : Rhombuses

Example Question #403 : Advanced Geometry

Example Question #111 : Plane Geometry

Example Question #112 : Plane Geometry

Example Question #21 : Rhombuses

Example Question #21 : How To Find The Area Of A Rhombus

Example Question #115 : Plane Geometry

Example Question #31 : How To Find The Area Of A Rhombus

Example Question #32 : Rhombuses

Example Question #33 : Rhombuses

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