Find the perimeter of triangle ABC using the 30-60-90 triangle rules. Simplify your answer as much as possible. ### Diagram of a 30-60-90 Triangle#### Description:The image depicts a 30-60-90 triangle, which is a special type of right triangle. The angles of the triangle are labeled, and one of the sides is provided.#### Components:1. **Vertices**: - **A** (Top vertex where the two equal sides meet) - **B** (Bottom right vertex) - **C** (Bottom left vertex)2. **Angles**: - ∠CAB (Angle at vertex C) = 60° - ∠ABC (Angle at vertex B) = 30° - ∠BAC (Right angle at vertex A) = 90°3. **Sides**: - Side opposite the 30° angle (AB) is labeled as 8 units long. - Side opposite the 60° angle (AC) is not labeled. - The hypotenuse (BC), opposite the 90° angle, is unlabeled.#### Explanation of the Triangle Properties:1. In a 30-60-90 triangle, the sides have a consistent ratio. - The length of the side opposite the 30° angle is half the length of the hypotenuse. - The length of the side opposite the 60° angle is $ \sqrt{3} $ times the length of the side opposite the 30° angle. - If one side is known, the other sides can be calculated using these ratios.2. **Given**: Side opposite the 30° angle (AB) = 8 units. - The hypotenuse (BC) = 16 units (twice the length of AB). - The side opposite the 60° angle (AC) = $8 \sqrt{3}$ units.#### Conclusion:Understanding the properties of a 30-60-90 triangle allows for quick calculations of side lengths based on one known side. This type of triangle is frequently encountered in geometry and trigonometry, making it an essential topic in mathematics education.

Answered: A 60° 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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Find the perimeter of triangle ABC using the 30-60-90 triangle rules. Simplify your answer as much as possible.

### Diagram of a 30-60-90 Triangle

#### Description:
The image depicts a 30-60-90 triangle, which is a special type of right triangle. The angles of the triangle are labeled, and one of the sides is provided.

#### Components:
1. **Vertices**:
- **A** (Top vertex where the two equal sides meet)
- **B** (Bottom right vertex)
- **C** (Bottom left vertex)

2. **Angles**:
- ∠CAB (Angle at vertex C) = 60°
- ∠ABC (Angle at vertex B) = 30°
- ∠BAC (Right angle at vertex A) = 90°

3. **Sides**:
- Side opposite the 30° angle (AB) is labeled as 8 units long.
- Side opposite the 60° angle (AC) is not labeled.
- The hypotenuse (BC), opposite the 90° angle, is unlabeled.

#### Explanation of the Triangle Properties:
1. In a 30-60-90 triangle, the sides have a consistent ratio.
- The length of the side opposite the 30° angle is half the length of the hypotenuse.
- The length of the side opposite the 60° angle is $ \sqrt{3} $ times the length of the side opposite the 30° angle.
- If one side is known, the other sides can be calculated using these ratios.

2. **Given**: Side opposite the 30° angle (AB) = 8 units.
- The hypotenuse (BC) = 16 units (twice the length of AB).
- The side opposite the 60° angle (AC) = $8 \sqrt{3}$ units.

#### Conclusion:
Understanding the properties of a 30-60-90 triangle allows for quick calculations of side lengths based on one known side. This type of triangle is frequently encountered in geometry and trigonometry, making it an essential topic in mathematics education.

Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

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