Find the PERIMETER of the following RECTANGLE. Simplify your answer as much as possible. ### Understanding Special Right TrianglesIn 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.

Name: Find the PERIMETER of the following RECTANGLE. Simplify your answer a
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Description: Find the PERIMETER of the following RECTANGLE. Simplify your answer as much as possible. ### Understanding Special Right TrianglesIn this educational example, we examine a rectangle with a diagonal line segment that cuts through it, forming two righ

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Answered: 30° 10 cm 30°

Math

Geometry

30° 10 cm 30°

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.

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