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
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Question
Find the PERIMETER of the following RECTANGLE. Simplify your answer as much as possible.
![### Understanding Special Right Triangles
In this educational example, we examine a rectangle with a diagonal line segment that cuts through it, forming two right triangles. The diagram provides details of the internal angles and one side measurement.
#### Diagram Description:
- The shape given is a rectangle.
- A diagonal splits the rectangle into two congruent right triangles.
- Each acute angle adjacent to the diagonal measures \(30^\circ\).
- The diagonal has a length of 10 cm.
#### Characteristics:
1. **Angles:**
- Each triangle formed within the rectangle has three angles: one at \(90^\circ\) from the rectangle, and the other two acute angles each measuring \(30^\circ\).
- This creates two \(30^\circ\)-\(60^\circ\)-\(90^\circ\) right triangles.
2. **Triangle Properties:**
- In a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle, the ratio of the lengths of the sides opposite these angles are \(1 : \sqrt{3} : 2\).
- The side opposite the \(30^\circ\) angle is half the length of the hypotenuse.
Given that the hypotenuse of the \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle (which is the diagonal of the rectangle) measures 10 cm:
- The side opposite the \(30^\circ\) angle is:
\[ \frac{10 \text{ cm}}{2} = 5 \text{ cm} \]
- The side opposite the \(60^\circ\) angle is:
\[ 5 \text{ cm} \times \sqrt{3} \approx 8.66 \text{ cm} \]
Thus, the base and height of the rectangle are 5 cm and approximately 8.66 cm, respectively. This shows how the diagonals and angles inside geometric figures help in calculating dimensions using trigonometric relationships.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff66317fc-ed86-4f22-9308-86904c21cc6d%2F4b092496-e5fb-4470-9495-70fd27ff5d73%2F78g3zym_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Special Right Triangles
In this educational example, we examine a rectangle with a diagonal line segment that cuts through it, forming two right triangles. The diagram provides details of the internal angles and one side measurement.
#### Diagram Description:
- The shape given is a rectangle.
- A diagonal splits the rectangle into two congruent right triangles.
- Each acute angle adjacent to the diagonal measures \(30^\circ\).
- The diagonal has a length of 10 cm.
#### Characteristics:
1. **Angles:**
- Each triangle formed within the rectangle has three angles: one at \(90^\circ\) from the rectangle, and the other two acute angles each measuring \(30^\circ\).
- This creates two \(30^\circ\)-\(60^\circ\)-\(90^\circ\) right triangles.
2. **Triangle Properties:**
- In a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle, the ratio of the lengths of the sides opposite these angles are \(1 : \sqrt{3} : 2\).
- The side opposite the \(30^\circ\) angle is half the length of the hypotenuse.
Given that the hypotenuse of the \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle (which is the diagonal of the rectangle) measures 10 cm:
- The side opposite the \(30^\circ\) angle is:
\[ \frac{10 \text{ cm}}{2} = 5 \text{ cm} \]
- The side opposite the \(60^\circ\) angle is:
\[ 5 \text{ cm} \times \sqrt{3} \approx 8.66 \text{ cm} \]
Thus, the base and height of the rectangle are 5 cm and approximately 8.66 cm, respectively. This shows how the diagonals and angles inside geometric figures help in calculating dimensions using trigonometric relationships.
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