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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![### Understanding the Geometry of a 30°-60°-90° Triangle
In the image, there is a triangular section RG with a bridge in the background, marked clearly to form a right triangle. Points R, B, and G denote the vertices of the triangle, with a 90° angle at vertex B.
The triangle BRG is specified as a 30°-60°-90° triangle:
- The hypotenuse (BRG) = 28 feet
- \(\angle B = 30^\circ\)
#### Problem:
What is the length of \( RG \)?
#### Solution:
In a 30°-60°-90° triangle, the sides have specific ratios:
- The side opposite the 30° angle (shortest side) is \( \frac{1}{2} \) of the hypotenuse.
- The side opposite the 60° angle (longer leg) is \( \frac{\sqrt{3}}{2} \) of the hypotenuse.
- Hypotenuse = 28 feet
To find \( RG \) (the longer leg), use the ratio for the 30°-60°-90° triangle:
\[ RG = \frac{\sqrt{3}}{2} \times 28 \]
#### Calculation:
\[ RG = 14\sqrt{3} \text{ feet} \]
#### Answer:
\[ \boxed{14\sqrt{3}\,\text{ft}}\]
#### Options Given:
- \( 14\, \text{ft} \)
- \( 16\, \text{ft} \)
- \( 14\sqrt{2}\,\text{ft} \)
- \( 14\sqrt{3}\,\text{ft} \)
#### Explanation of Diagram:
The diagram displays a triangular section of the bridge with:
- Point R located at the top and to the right.
- Point B located at the bottom left, forming a right angle with the base and height.
- Point G forming the hypotenuse at the right top corner.
The yellow lines trace the right triangle accurately on the bridge structure.
By utilizing the properties of a 30°-60°-90° triangle, we can solve for the unknown side "RG," and according to the ratio given, the answer is \( 14\sqrt{3}\,\text{ft}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdfa3a23b-5b1c-495e-9acf-5b0fca583888%2F47680796-dad0-4d93-8806-e3e683c4128a%2Fa5jwk18_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding the Geometry of a 30°-60°-90° Triangle
In the image, there is a triangular section RG with a bridge in the background, marked clearly to form a right triangle. Points R, B, and G denote the vertices of the triangle, with a 90° angle at vertex B.
The triangle BRG is specified as a 30°-60°-90° triangle:
- The hypotenuse (BRG) = 28 feet
- \(\angle B = 30^\circ\)
#### Problem:
What is the length of \( RG \)?
#### Solution:
In a 30°-60°-90° triangle, the sides have specific ratios:
- The side opposite the 30° angle (shortest side) is \( \frac{1}{2} \) of the hypotenuse.
- The side opposite the 60° angle (longer leg) is \( \frac{\sqrt{3}}{2} \) of the hypotenuse.
- Hypotenuse = 28 feet
To find \( RG \) (the longer leg), use the ratio for the 30°-60°-90° triangle:
\[ RG = \frac{\sqrt{3}}{2} \times 28 \]
#### Calculation:
\[ RG = 14\sqrt{3} \text{ feet} \]
#### Answer:
\[ \boxed{14\sqrt{3}\,\text{ft}}\]
#### Options Given:
- \( 14\, \text{ft} \)
- \( 16\, \text{ft} \)
- \( 14\sqrt{2}\,\text{ft} \)
- \( 14\sqrt{3}\,\text{ft} \)
#### Explanation of Diagram:
The diagram displays a triangular section of the bridge with:
- Point R located at the top and to the right.
- Point B located at the bottom left, forming a right angle with the base and height.
- Point G forming the hypotenuse at the right top corner.
The yellow lines trace the right triangle accurately on the bridge structure.
By utilizing the properties of a 30°-60°-90° triangle, we can solve for the unknown side "RG," and according to the ratio given, the answer is \( 14\sqrt{3}\,\text{ft}\).
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