Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
Related questions
Question
![### Problem 2: Find the Area of the Rhombus
#### Description:
The problem presents a rhombus and asks to find its area. The rhombus is displayed with the following details:
- One side length is given as 12 units.
- An internal angle of \(60^\circ\) is marked.
- Diagonals are represented by dashed lines, and their intersection forms four right angles, indicating that they bisect each other perpendicularly.
#### Diagram:
- The rhombus is grey-shaded.
- The mark \(60^\circ\) is shown between two sides of the rhombus, indicating one of its angles.
- Both diagonals and side lengths are marked with distinguishing lines to indicate their equal lengths.
#### Solution:
To find the area of the rhombus, you can use the formula involving the side length \(a\) and one of the internal angles \(\theta\):
\[ \text{Area} = a^2 \sin(\theta) \]
Given:
\[ a = 12 \, \text{units} \]
\[ \theta = 60^\circ \]
Substituting the values:
\[ \text{Area} = 12^2 \sin(60^\circ) \]
\[ \text{Area} = 144 \times \left(\frac{\sqrt{3}}{2}\right) \]
\[ \text{Area} = 144 \times 0.866 \]
\[ \text{Area} = 124.70 \, \text{square units} \]
Thus, the area of the rhombus is approximately \(124.70\) square units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb602927-d5c2-4b17-99bc-cf43c066f5a5%2F9b774de8-8440-40a8-9847-a35b37bd5cf6%2F743uodn_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 2: Find the Area of the Rhombus
#### Description:
The problem presents a rhombus and asks to find its area. The rhombus is displayed with the following details:
- One side length is given as 12 units.
- An internal angle of \(60^\circ\) is marked.
- Diagonals are represented by dashed lines, and their intersection forms four right angles, indicating that they bisect each other perpendicularly.
#### Diagram:
- The rhombus is grey-shaded.
- The mark \(60^\circ\) is shown between two sides of the rhombus, indicating one of its angles.
- Both diagonals and side lengths are marked with distinguishing lines to indicate their equal lengths.
#### Solution:
To find the area of the rhombus, you can use the formula involving the side length \(a\) and one of the internal angles \(\theta\):
\[ \text{Area} = a^2 \sin(\theta) \]
Given:
\[ a = 12 \, \text{units} \]
\[ \theta = 60^\circ \]
Substituting the values:
\[ \text{Area} = 12^2 \sin(60^\circ) \]
\[ \text{Area} = 144 \times \left(\frac{\sqrt{3}}{2}\right) \]
\[ \text{Area} = 144 \times 0.866 \]
\[ \text{Area} = 124.70 \, \text{square units} \]
Thus, the area of the rhombus is approximately \(124.70\) square units.
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